Had this problem playing Halo: MCC and a few other games, taking Screenshots would freeze the game for up to a second and this would desync voice acting from the game animation making it look weird and also propagate to my CoOp partner. When this also appeared in various of the recent Demos (kind of inconsistently) and Detroit: Become Human I realized this isn't a game-specific problem, but has to do with how Steam handles Screenshots. In my case the biggest problem was a folder having ~15k files.

After trying a bunch of things related to uncompressed Screenshots in the setting or just moving the large Screenshot folder out which did nothing, the solution described in another topic seems to have worked: https://steamhost.cn/steamcommunity_com/discussions/forum/1/3145094199288776388/

First I moved the above folder out so that Steam doesn't recognize it anymore, then I renamed "screenshots.vdf" and let Steam rebuild it upon a restart (it didn't rebuild it for some games my account has no licenses for anymore like expired Demos though).
The result was no more lag/stutter when taking Screenshots during Detroit: Become Human anymore.

A likely more permanent solution to this would be if Steam changed how it handled Screenshots and for instance didn't add new Screenshots immediately to the .vdf file when they are taken, but in the background after a gaming session is over.

Despite the most annoying part of this being resolved by doing the above, opening the new Screenshot Manager or trying to browse down too quickly while on "Most Recent" and not any specific game sometimes makes it crash with a reactjs error like this, since there are still a lot of Screenshot files remaining for other games, likely also related to the amount of them:

Something went wrong while displaying this content. Refresh
Error Reference: Shared SteamUI_8145392_f385dbfebf120acd
Minified React error #185; visit https://reactjs.org/docs/error-decoder.html?invariant=185 for the full message or use the non-minified dev environment for full errors and additional helpful warnings.