Vlad — Yup, just tried in a brand new profile. I entered one of the links I had an issue with yesterday and it did eventually load, but it hung at a blank page for like 2 minutes before loading, and then when the page was loaded there were some weird page styling issues that I don't see when using other browsers (eg some elements look like they are in dark mode, others are in light mode, floating buttons leave artifacts when scrolling).
I'm not sure if this is the exact same as the issue described in the original post as most sites are working for me, but some won't load seemingly at random and in the exact same way as shown in the videos above.
Here's an example of a link that does what I've described on my machine, maybe try it on yours and see if it behaves normally: https://forums.macrumors.com/threads/apple-violated-u-s-labor-laws-with-anti-leak-email.2378931/post-31925349