update software

Catalina is 4 years old ,so there are good chances that your safari is indeed outdated.

If you cannot update further due to the Age of your Mac, I suggest that you're going to use a third party browser, like Firefox. That one still gets updates for Catalina

Your Mac being updated to its latest macOS version available does not mean it meets the required security of any given website. The website may be requiring security your Mac is not capable of providing. Your Mac is a full four versions behind even if it's at its max OS allowed.

Try the latest version of Firefox or Brave browser.

The Safari browser sometimes gets its own update, but typically is updated with a macOS update or upgrade. Of course, that only happens if your Mac can support the newer macOS version. But you may not actually have to upgrade the macOS to get your website access back.

If you are having trouble connecting to and using websites with your older version of Safari, try it with a newer third party browser that is still being updated to meet current internet security protocols.

I suggest you take a look at Firefox browser or Brave browser... or even Chrome 🙄.

Download an alternative browser and install it into your Applications folder. Launch and use that when you encounter websites that don't play well with Safari.

Was the website that told you Safari needed to be updated a Chase bank website, by any chance? Chase is the only financial institute I've read about that's creating this issues for their users.

If you can't update your Mac beyond where it is now, then your only alternative is to use an alternative web browser, as D.I Johnson mentioned.