When you update your OS will it automatically update your Safari browser

Safari 16 requires Big Sur (macOS 11), Monterey (macOS 12) or later. It is not available for Catalina 10.15.x.

Safari 17 requires Monterey (macOS 12) or later.

In addition, in my experience Safari updates have always been individual updates, not part of a macOS update.

Did you say that you're running Catalina on your Mac? The current version of Firefox will run on Catalina. It might help with some sites that are giving you trouble with Catalina's version of Safari.

https://www.mozilla.org/en-US/firefox/new/

Yes, but I would go to the app store and check just to be sure

Internet security features continue to evolve and older versions of Safari cannot negotiate secure connections with newer sites.

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. 

And, yes, the Safari browser is typically updated with a macOS update or upgrade. Of course, that only happens if your Mac can support the newer macOS version.

An OS update updates all native Apple apps if the dataset contains new app updates. Not every app is updated every time, but the mechanism to do so is the OS update.

There are sometimes individual updates, notably Safari.

You're welcome. 👍🏽

Reboots are good, too, and should be done at lease once a week.