Ok! I was wondering about that too. My laptop can't upgrade to Sonoma, but on Ventura, we have been getting updates to Safari. The most notable change was the addition of Profiles. Ever since that update, Safari's performance has been very uneven. Sometimes it runs normal, and sometimes it would just hang on particular webpages. I was thinking it may have something to do with iCloud private relay, which I do use, but maybe that's not it.
Because of these performance issues, I no longer use Safari as my main browser for work, which is sad. I much prefer it for a number of reasons including privacy and that it crosses all my personal devices.
Sounds like Private Relay. I have it turned off on my home network as I’ve noticed issues similar to your’s with it enabled.