I still have High Sierra and am wondering if there is any way to upgrade this, short of buying a new computer!

Back up your data, check your applications for compatibility, click here, and if your MacBook Pro is a 2012 or newer model, then on the download link for a newer OS.

Catalina and higher won't run 32-bit software; download and use Go64 as well as using the first link above.

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And if you can't...

Mostly you just need a more modern Browser than Safari on High Sierra like Firefox or Brave.

https://brave.com/