The most important question. ;)

There are hints it might happen:
https://www.gamingonlinux.com/2022/04/turbo-overkill-should-see-full-linux-and-steam-deck-support-for-the-10-release/

Is anybody here who knows more? The game looks fun, but I'm tired of spending money for games that are W32 only.

I assume it will be playable thanks to Proton on Linux systems, like most of the games nowadays. I think it will take some time until we see more native Linux games.

I just moved to Linux this year and Turbo Overkill is the only game in my entire library giving me any trouble at this point, but I don't believe it's a Proton issue. I've had every game I've tried running on different proton versions testing them out and getting my feet wet and since about 8.13 this is the only game I have any trouble with, Seems to load to the menu but it will not go fullscreen and if I can trick it into that it's just a back screen that won't minimize or tab out. I can hear the music though. Very frustrating.
Again I don't think it's a Linux or Proton issue, but it's worth mentioning that this is the only game out of at least 100 tested that is giving me problems, from MDK to Doom Eternal to ePSXe launched through the bignamelauncher using Protontricks, Turbo Overkill is the only thing giving me trouble on this system.

On Linux I install my GOG games with the Heroic Launcher. Earlier today I was able to launch the demo version with Proton 7.0. Only downside are some missing cinematics (parts of the intro, tutorial videos), apart from that the game worked fine.

I've been using Heroic primarily and couldn't like it more, works great. I have tried Proton 7.0 for Turbo Overkill as it's certified for that but no luck. Tried a few command line arguments to force screens but no luck there either.

Do you have "Use Steam runtime" ticked in the settings of the game in Heroic?

Didn't make a difference, dragging the executable onto prefix in Heroic, running the demo, installing via wine myself, it all goes as it should but results in a permanently minimized black screen regardless of what libraries I use or how I install and run it. I think it's a broader system issue because many, but not all or even most, but many full screen programs begin minimized on this system, though most just tab up to full screen no problem. I'll figure it out at some point and post results if the thread's still around.

Interesting, what distro do you use? I am usually on Debian-based ones, currenty on Zorin OS.

Currently on Mint 21.2 Victoria, Ubuntu 22.4 based.

I have just tried to launch the demo on a pretty fresh install of Mint 21.2 and the problem with in-game videos is resolved by using the Proton-GE-latest version. Only one issue occurs, on my widescreen in older Proton versions there were black bars on the left and right side of the screen, now the game uses all the space of my screen and the buttons in the menu work only when I get out of fullscreen mode with ALT+ENTER and get in fullscreen again. It's like the game thinks the menu buttons are in the positions when the game launches with the black bars, which is no longer the case.

That's a little different from my exact issue but seems to still touch on the whole issue of not wanting to just run normally in fullscreen. This system in particular seems to really love opening fullscreen games minimized and I think when I find that gremlin I'll be able to sort this out. Thank you for sharing all the info.

Update:

Fresh install of Mint 21.2, running through Heroic games launcher with ProtonGE latest, 8-22 at the moment.

Intro videos don't render and I had to alt-tab into the menu but it showed up and ran fine, played the first fight no problem.

Gets a big thumbs up from me!