Have just ran into the same issue after upgrading to Ventura 13.0 a couple of days ago and tearing my hair out trying to troubleshoot. I'm a web developer so I could see the errors in the browser console but didn't know what was the root cause, other than I was suddenly having problems across multiple websites immediately after upgrading. Firefox was completely unaffected.
The biggest issue I saw was that Google Captchas were not loading at all which meant that any form that used them, whether it be registration, login, and even many newsletter signup forms would no longer validate and thus no longer submit. The worst example I saw was a website which had a newsletter signup in the website footer with a Captcha and because of this the javascript on every webpage was broken as it was waiting for the Captcha to load which would never happen. Toggling the "Hide my IP address from trackers" would get the Captcha to appear and the webpage to function as normal.
I did clear my browser cache which seemed to have no affect. I also "Enabled Intelligent Tracking Prevention Debug Mode" but this didn't seem to give any clues or warnings.
Given how widely used Google Captchas are, and "Hide my IP address from trackers" is enabled by default, there is a bug somewhere which is only affecting a few of us for some reason. I don't know what that is. Perhaps cookie related for previously visited websites which then are causing an issue after the OS upgrade?
Hopefully this can help someone else out or find the route cause.