If web browsing with Safari is the main or only problem, download & install Firefox. It is fully supported on Catalina and has very good security & privacy features. It should work fine with the websites you say don't currently load (assuming you are using Safari).
Regarding other apps or issues, your late 2012 27 inch iMac can run Catalina but not more recent versions of macOS. If you want to be completely up to date you will need to replace your iMac with a newer model. And at this point it should be an M-series (Apple Silicon) Mac, not an Intel Mac.