How to turn off autofill making the fields pop out?

For anybody who does not like animations, you can disable Bitwarden’s autofill animations by disabling animations system-wide.

On Windows 11, go to Start > Settings > Accessibility > Visual Effects , and disable the option “Animation Effects”. This will suppress Bitwarden’s animations (and other animations).

I would be interested to hear whether this approach can be used in other operating systems

If you don’t want to disable animations system-wide, then the only recourse for now is to support the relevant feature request: