Often times, games that have a native Linux runtime will still play better using proton and the Windows version. So try that with cs2. Only source game I've played is HL2 (recently) and i think I might have had to switch to Proton for it to run well, but I forget.

There isn't really anything wrong to do... The first time you load up a game, it might take a little bit longer, so you might have to let it sit for a minute while Steam says "exit" instead of "play game" (meaning the button you click in Steam to start the game). That's only the first load, after that, it should load right up.

You might also want to look at you settings for loading shaders (steam settings). I prefer to compile shaders before the game starts so as to avoid stuttering in the first few min, but it can also make it so it takes an additional 30 sec or so to open the game.

Another suggestion I'd have is to get proton-GE. Basically it's some dude's custom proton release that usually has better compatibility and often works better than official (if you want to avoid having to download a separate program, try "proton experimental" in your Steam compatibility options).

But getting proton-GE is easy. You have to switch to desktop mode once and use Discover (app store basically) to download "protonupQT" (I believe, this is all from memory so it might be slightly off). You can run that app in desktop mode, or do what I do and go back to game mode and add it as a "non-Steam game". Now you can update proton-ge without having to go into desktop mode...

Anyway, I'm sure that seems like a lot but it's really not. Hope it helps.

Edit: I'm just realizing now that your didn't mean "Counter Strike," but rather "Cities Skylines". Yeah, can't speak to the performance there. Still worth trying Windows version with proton