In addition to suggestions from other answers, here are some additional points.
Evaluating The Biases From Different Sources
The Oryx blog estimate, is a conservative lower bound because its confirmation requirement necessarily means that there are other unconfirmed losses that also exist and there is no estimation of what percentage of losses are unconfirmed. But, since its standards are consistent for all sides, the relative magnitude of each side's losses should be fairly accurate.
It is also fair to view Ukraine's reports of Russian losses as an upper bound on possible Russian losses, and to view Russian reports of Ukrainian losses as an upper bound on Ukrainian losses.
For example, the Russian News agency TASS claimed on February 28, 2022 that:
Russian troops destroyed 1,146 Ukrainian military infrastructure
facilities during their special operation for demilitarizing Ukraine,
Russian Defense Ministry Spokesman Major-General Igor Konashenkov said
"Since the beginning of the special operation, the Russian Armed
Forces have crippled 1,146 Ukrainian military infrastructure
facilities, including 31 command and communications centers, 81 S-300,
Buk-M1, and Osa surface-to-air missile systems, and also 75 radar
stations," the spokesman said.
Operational-tactical aircraft of Russia’s Aerospace Force wiped out
six armored columns of the Ukrainian armed forces, the general said.
The Russian combat planes "destroyed 311 tanks and other combat
armored vehicles, 42 aircraft and helicopters (including those on the
ground), 51 multiple launch rocket systems, 147 field artillery guns
and mortars, and 263 special military motor vehicles," the Defense
Ministry spokesman said.
While it is possible (and indeed likely) that Russian forces destroyed less Ukrainian military equipment than that in the first five days of the conflict, it is almost certain that it did not destroy more than that much Ukrainian military equipment by then. It does appear, however, from independent news reports, that almost all fixed Ukrainian radar stations were destroyed in the first day or two of the conflict.
Ukraine has not publicly reported on any of its own military losses, to the best of my knowledge, although it has reported its own civilian losses.
Russia has made reports of its own military losses, but these reports (of perhaps 500 Russian soldiers killed) are absurdly low compared to any other reasonable estimates from other sources.
The U.S. Pentagon numbers are informed by satellite images in real time and other sources of intelligence of a variety of means. Also, importantly, unlike the Oryx blog, the U.S. Pentagon makes at least a rudimentary effort to estimate the total losses from incomplete information.
This is probably the closest to accurate data available in absolute terms. So, it can be used to some extent to calibrate data like that from the Oryx blog, except that it is much less detailed in its assessments of equipment losses, instead focusing on lost lives and injuries.
Estimates of equipment losses provide some guidance regarding lives lost, injuries, and captured or deserting soldiers, but only a crude one. Ideally, one would look at changes in Oryx blog, Ukrainian government, and Russian government claims over time and compare them to changes in the Pentagon estimates over time, in an effort to calibrate this kind of estimate.
Other Estimation Tools
Expected Russian v. Ukrainian Losses
As of March 10, 2022 (a week old, and hence stale):
The [Oryx] website has counted significantly more Russian equipment losses
than Ukrainian losses: 993 versus 277. Of particular importance, it
has counted losses of 465 Russian heavy vehicles (tanks, armored
fighting vehicles, infantry fighting vehicles, armored personnel
carriers), versus 138 for the Ukrainians.
Forbes magazine offers another rule of thumb from military experts looking at past conflicts:
An invading army should expect three times the losses of the army on
the defense, if the two armies are equally matched. These numbers
suggest that the two militaries are fairly equally matched and that
neither side necessarily has a technology advantage.
Deaths v. Combat Injuries
The estimate that two or three soldiers have suffer serious injuries, but are not killed for every soldier killed in modern warfare, consistent with the U.S. Pentagon's most recent estimates of Russian military casualties, is likely a common rule of thumb based upon experience in previous military conflicts.
Reality Checks From The Number Of Russian Generals Killed
Another relevant point is that there are credible reports that at least four Russian Army generals have been killed in the conflict. Multiple sources have also reported that something on the order of 180,000 to 200,000 Russian troops have been deployed in Ukraine. In the U.S. Army, there is about one general per 2,000 U.S. Army soldiers and the ratio is very similar in the U.S. Marine Corps. This ratio would be not all that greatly different in the Russian Army, so there are probably something on the order of 90-100 Russian Army generals involved in the Ukraine conflict.
There has been no military conflict since the early 19th century, at least, where Army generals died at a higher rate than rank and file Army soldiers in active combat (ignoring post-defeat retribution directed at surrendered or captured generals). So, it is safe to infer that it is highly likely that more than 4% of rank and file Russian troops are casualties.
Small numbers, like four general officer casualties, aren't a very precise estimate. Maybe the Ukrainians just got lucky and killed a disproportionate number of generals by sheer chance.
But this number does tend to corroborate the U.S. Pentagon estimate that 7,000 Russian troops have been killed (roughly 3.5%), and that another 14,000-21,000 Russian troops have been seriously injured (roughly 7-10%) for a combined casualty rate including soldiers both killed or injured (as well as soldiers who are captured or have deserted) something on the order of 10-14% of the Russian force.
Indeed, if anything, the fact that four Russian generals have been killed suggests that this 10-14% total casualty rate estimate for Russian troops is conservative and is probably on the low end of the actual total.
Other Inferences About Russian Military Resources
One can also make some more qualitative or crude estimates based upon the kind of tactics that are being employed by the respective sides in the conflict.
Guided Missiles and "Smart Bombs"
For example, there are strong indications that after the initial barrage of missiles from Russian forces on February 24, 2022, that Russian forces have made little or no use of guided missiles or "smart bombs", even though it has at least some of these in it national arsenal.
Guided weapons are, however, vastly superior.
For example, a U.S. guided multiple rocket launcher missile (typically considered a form of artillery) can hit a target with a 10-20 foot (3–6 meter) target radius at a distance of up to about 36 miles (60 km). In contrast, an unguided mobile howitzer artillery shell has a 900 foot (300 meter) target radius (i.e. about 2000 times larger an area where it is 50% likely to hit) at its maximum range of about 18 miles (30 km) and a 180 foot (55 meter) target radius (i.e. about 80 times larger an area where it is 50% likely to hit) at its optimum range. Both deliver explosive effects that are in the same general ballpark in magnitude.
So, it is reasonable to infer that if Russia had guided missiles left, it would be using them. Yet, the news reports from the conflict indicate that Russia is overwhelmingly shelling its opponents with conventional howitzer artillery rounds and unguided rockets, rather than using guided missiles.
A similar inference can be made regarding aircraft delivered bombs. A guided "smart" bomb is about 300 times more likely to hit its intended target than a 1970s-1980s era unguided aircraft dropped bomb. And, a "smart bomb" can be dropped from a higher altitude where it is less exposed to surface to air anti-aircraft missiles and anti-aircraft gun fire, which is a big concern because news reports have established that Western allies of Ukraine have supplied it with at least hundreds, if not thousands, of small, "man carried" Stringer anti-aircraft missiles and the equivalent.
Yet, news reports have also established that Russian aircraft making bombing runs in Ukraine have been using old school "dumb bombs".
It is fair to infer from this fact that Russian forces have pretty much completely exhausted their supply of "smart bombs", given, the greatly increased danger to Russian warplanes and pilots involved in using dumb bombs when "smart bombs" are available, and the fact that "smart bombs" allow bombing aircraft to make vastly fewer bombing runs in an environment where the number of bombing runs available to Russian forces is limited in an environment where it has not have air superiority.
Black Sea Fleet Involvement
We can also infer that the Russian Navy's Black Sea fleet, which consists of 45 ships (not all major surface combatants) and seven submarines, have had little or no involvement in the ongoing conflict after the initial February 24, 2022 barrage of missiles fired across Ukraine, many of which were guided missiles.
There are no reports of naval battles, which surely would have been reported if they occurred, either by the winning side's military, or by the U.S. Pentagon which would have observed this with their satellites.
Also, Russian warships have only two kinds of weapons that can be directed at ground targets: guided missiles, which don't appear to have been used in any significant number since the initial day of the conflict three weeks ago, and naval guns which are basically big unguided shell firing howitzers mounted on ships. But, naval guns have a maximum range of 12-24 miles (20-40 km) depending upon size, only a quite small portion of the shelling in the conflict has taken place close enough to the shore for that option to even be possible, and there are no reports of this being done.
The possibility of Russian Navy fire support for ground troops is also a disfavored possibility because the technical details of coordinating target locations between warships and ground forces are great and there are few, if any, suggestions prior to the conflict that the Russian Navy and Army had developed this capacity to coordinate).
Estimating Percentage Losses
It is also worth noting a couple of important issue comes up with trying to compare the losses to the side of the Russian and Ukrainian forces immediately prior to February 24, 2022, in order to determine losses on a percentage basis.
First, there is considerable disparity between sources over, for example, how many tanks and troops each side's army had prior to the conflict. The Forbes article linked above, for example, reports that:
Russia went into the war with 9,780 tanks while Ukraine had 2,170.
But, the World Almanac 2022 (I rely on the printed version here) citing The Military Balance, 2021 prepared by the International Institute for Strategic Studies, reported that as of November 2020, the Russian Army had 2,840 main battle tanks in its entire Army worldwide, while the Ukrainian Army had 858 main battle tanks.
In this case, the IISS number is almost surely more accurate, because I have seen numerous media reports over the last decade or two reporting that Russia has retired many thousand of its main battle tanks to cold storage for parts and the like, from a peak number close to the one cited by Forbes.
Since there is no good reason to believe that the Russian Army has deployed every single one of its operational and in service main battle tanks to the Ukraine theater in this conflict, moreover, the number of tanks that the Russian Army has deployed is surely significantly less than 2,840 (even adjusting for forces that are part of the Army of Belorussia deployed in this action).
The IISS reports that Russia, worldwide, has 900,000 active duty soldiers and sailors (including at least 150,000 sailors in the Russian Navy), 2,000,000 reserve soldiers and sailors, 185 air to air fighter aircraft, and 415 ground attack fixed wing fighter aircraft as of November 2020 (2021 for soldiers and sailors), while the IISS reports that Ukraine has 209,000 active soldiers and sailors, 900,000 in reserves (the reserve figure is higher than estimated in recent media reports), 71 air to air fighter aircraft, and 14 ground attack fixed wing fighter aircraft.
One reasonable tool to estimate the number of tanks deployed in Ukraine is to assume that if the number of Russian soldiers involved is about 25% of its active duty force outside the Navy, which suggests that something on the order of 620 tanks have been committed by Russia to the Ukraine war.
Given the Oryx and Ukrainian military estimates cited above, this suggests that at least 1/3rd and something less than 2/3rds of Russian tanks deployed to Ukraine have been destroyed by the military of Ukraine.
Second, while essentially 100% of Ukrainian military resources are in play in this conflict, a significant share of Russian and Belorussian military resources are deployed outside the theater of conflict, for example, in Syria and in military bases near the Baltic states.
Third, while the available supply of military equipment on the Russian side of the conflict is more or less fixed, because it hasn't had international allies supplying it with military equipment over the last three weeks, Ukraine has been receiving a substantial volume of lightweight military equipment resources from its Western allies, such as something on the order of ten thousand anti-tank missiles and at least hundreds of drones. This doesn't appear to have included significant numbers of tanks, manned warplanes and helicopters, or other manned military vehicles, however.
In the other direction, Ukraine's self-report that is destroyed the sole frigate, which was the most capable warship in its Black Sea Navy, to prevent it from being seized by the Russians, supported by video images, seems credible.
Still, even with these limitations, one can comfortably say that even the Russian estimates are not consistent with the possibility that the entire Ukrainian air force has been destroyed or rendered useless by Russian invaders, which means that Russia, after three weeks, still does not control Ukrainian airspace.