We know why the West hates Assad, but, Turkey's hate is not clear to me.
The same reason, in fact. Turkey was among the primary sponsors of the Syrian war from the very beginning. The USA, UK, France, Saudi Arabia, Qatar and Turkey - this is the short list of the states which started all this mess in 2011. Moreover, it was nearly impossible to do without Turkish assistance - just look at the map.
So Erdogan didn't start the war, because he hates Assad, but, vice versa, he hates Assad, because he started the war against him. Of course, now the situation has changed in political and military sense, but Erdogan has no reason to make friends with Assad anew (yes, they were on quite good terms before all that happened). The point is that Assad is under the tight control of Iran and Russia and has a little value of his own.
So when Erdogan speaks for "western" or national press, he says that Assad is "a public enemy" and such, but when he speaks with Russians, he has absolutely no problems with cooperating with Damascus. In fact, his doublethink is quite amusing.
UPD. on 04.06.19:
I am curious as to their motivations and why they are involved, which I don't think you have (clearly) explained
That's an interesting matter, though it's only partially related to the original question.
Let's put it like this: the so-called "Arab Spring" is the great occasion and opportunity for many countries to get some big and yummy piece of the geopolitical pie. Yet it's natural to suppose that the primary initiators of the Syrian war were the same countries as with Iraq, Afghanistan, Yemen, and, possibly, forthcoming Iranian war. That is the united states of "America and Gulf" (and, yes, I mean all these wars should be viewed in common as a sort of "the New Great Game" in the Middle East).
But they certainly needed some allies: France, GB, Jordan, and, of course, Turkey - a large neighbour country and the member of NATO (there's also an interesting question about the role of Israel, but it became apparent only later, so I can't claim they had much to do with this in the first period of war; of course, there are other countries involved in it, but for such as Bulgaria, Czech Republic, Croatia, Germany, etc. this is mostly arms manufacturing/sales/smuggling business).
No one knows what exactly was promised to Erdogan, but during 2011-13 the anti-Syrian alliance sailed in the same boat. However, the most interesting events took place in 2014 when the Qatar-sponsored ISIS made a spectacular rise out of nowhere. They quickly occupied large territories in Iraq and Syria, and started to push on Saudi-sponsored Al-Qaeda/An-Nusra, USA-sponsored YPG, etc.
Turkey, being closely tied with Qatar, was accused in helping to ISIS many times by different parties (including Russia and USA). It's especially obvious that Turkey was among ISIS' most valuable trade partners in this time.
Then in 2015 the USA-controlled Kurdish guerillas broke the truce with Turkish government and carried out numerous terroristic acts during 2015 and 2016 resulting in the new round of the long-standing Kurdish-Turkish conflict.
The next attempt to solve "Turkish problem" was on 15-16 July 2016. Immediately after that Erdogan finally switched sides and started closely cooperate with the Russian-Iranian-Syrian alliance.
So what's next? ISIS seriously got under fire in 2016, and especially in 2017, when Qatar was blockaded by its neighbouring countries, because of "the alleged support of terrorism". And the most loyal ally who helped to Qatar to withstand these events was, of course, Turkey.
Of course, there's much more awaiting for us: the war in Syria still continues, and also the war in Libya, where Turkey and Qatar allied with Italy to support GNA against LNA/Khaftar, who is, in turn, supported by France, Saudi Arabia and Egypt. And maybe there'll be some war in Iran, who knows?
Most certainly the politicians have many plans and ambitions. But we don't know exactly their motivations and aims. All we can do is to analyze "the visible events" and to follow them chronologically.