Whenever I open Western Media I see negative news about Western countries like US, Britain etc.
Well, in countries with more press freedom it's almost a given most government policies or the current state of affairs will be criticized by someone writing in the press. The exceptions, i.e. when there is near universal consensus on something (and just heaping praise), are pretty rare in the West.
In countries with more government control over the press (legal or just de facto), there won't be that much criticism of their own country/government or possibly even more broadly even of their (regional) allies etc.
For example, I see videos insulting/ making fun of the very president of USA, of gun violence, racism etc , or in UK, of stabbing incidents, some religious violence (something about Hindu vs muslim etc).
To be honest, I'm not sure what this refers to exactly. There's a lot of variety in Western media from humorous/irreventious to much more serious in tone and sometimes devoid of commentary. Certainly though, making fun of their own head of state isn't criminalized in Western countries as often as it is in some other parts of the world.
The example you've provided, which Joe W addresses in his answer, is hardly a criticism of "the West" nor is it exactly an illustration of your claim (immediately quoted above). Rather that example shows a [group of] traditional media outlets [all with the same owner] criticizing social media, in general terms, but the thrust seems to be about trustworthiness and "fake news". What you see there is somewhat like an editorial of the [Sinclair] media group. And yeah it is a bit cheesy and self-serving. But while they provide no data to back up their claims, there are some surveys which suggest that fake news more easily spreads on social media, or at least that news stories first posted on social media are more often fake.
The Sinclair group actually has a somewhat long tradition of such "must-run" segments. One additional point of confusion in the organization of US (local) TV is that local stations can be owned by
Sinclair (or whomever else) but carry a [dual] affiliation with another network/brand (like ABC or NBC)... even though the main/national station of the latter is not owned by Sinclair nor does it carry Sinclair's programming/segments that you can see on the local [affiliated] stations.
does there exist quantitative evidence to believe that Western Media is biased against itself?
Somewhat of an aside, about [some Western] media criticizing "itself", or more commonly their competition (of some sort) from the same part of the world: the received wisdom one could hear a decade or two ago was that British journalists much more often did this, i.e. criticize/attack each other, while US journalists didn't do this as often. On the other hand, with the massive rise in political opinion/talk shows on US cable television, the anchors of the latter have become common targets of criticism in the US media. And this phenomenon goes hand in hand with the increased political polarization the US sees.