why has safari stopped working on certain websites

You can run up to the latest version of macOS, Catalina, so I would say no, it's not necessary to replace it unless the drive is failing. I would upgrade to at least High Sierra.

How to upgrade to macOS High Sierra - Apple Support

Works fine here. Which version of Safari and macOS are you running?

You need to be a bit more specific. Which websites?

Since you give no indication of what Mac you have it's not possible to say.

What year?

You're welcome.