I have exactly the same issue.
No, it isn't because of keyboard (Samsung or GBoard or anything similar). No, it's not because "show recent pages" being enabled.
I have tried many different settings, but no avail, until I found this page with suggestion from Dave Horne above. Yes, it's very ridiculous that dark mode fixes this jumping issue.
To someone who doesn't understand the issue, let me explain:
Android 12, Samsung Galaxy S10+, latest software and all apps updated as of now (4/28/2022).
Google Search Widget on home screen.
Tap on the Google Search widget, the widget opens the search screen, keyboard automatically appears allowing me to type.
However, immediately after typing the first character, the keyboard disappears, the page refreshes itself. It seems the Google Search is being replaced by Chrome or whatever right after the first character. It happens very quickly, but quick enough to make the keyboard disappear, so it is very annoying because then I have to click on the search bar again to continue typing.
After that, as long as the search isn't being closed by the system (or being forced to close by user), it won't happen again.
I have another Samsung S10+ which surprisingly doesn't have this issue with dark mode DISABLED. I also have other Galaxy phones with Google Search widget but none has this issue. On the Samsung Galaxy S10+ that has this issue, previously (maybe a month ago),it worked just fine. I did reset app defaults on this phone about a month ago. So that might have caused the issue. I have yet to find out which specific setting for this.