Safari Browser Issues

Yes, absolutely. Just be sure to create a Time Machine backup first: Back up your files with Time Machine on Mac - Apple Support.

To upgrade your Mac, read How to download and install macOS - Apple Support. Scroll down to the Catalina 10.15 link, or whatever the latest macOS version that Mac can use.

There are a few caveats with Catalina, among them: What happened to iTunes? - Apple Support and the fact that Catalina will no longer be able to run old, 32-bit apps: 32-bit app compatibility with macOS High Sierra 10.13.4 and later - Apple Support. The latter is probably more important since the new Music app will transition your existing iTunes Library and you will lose nothing, whereas 32-bit apps are now considered so old Apple finally pulled the plug on any way of running them. That really ought not to be a surprise because Apple has been warning us for about fifteen years now.

But yes, Catalina will incorporate a much more recent version of Safari that is almost certain to work with those websites and others. And you can still use Brave, or Firefox.

If your MacBook Pro is still running High Sierra, its version of Safari is quite old by now. You can try using newer browser such as Brave or Firefox (•) but they are also on their last versions that will run on High Sierra.

https://brave.com

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

• Only Firefox ESR will run on that Mac.