Safari does not open many commercial websites

Safari already works with all websites.

I suspect there is more to your question though. Are you having trouble using Safari? If Safari on Mac doesn't open a webpage or isn't working as expected - Apple Support may be applicable.

TNGWB wrote:

I wish!

I often cannot run applications on other websites

What do you mean by "run applications on other websites"? Can you give examples of specific sites you're having trouble with?

The last time I remember having a problem with a website and Safari, it was my state's unemployment benefits site (which appeared to date back to the beginning of the world wide web) and that was in 2016.

Run applications... I don't understand. If Safari is unable to load websites then then the Apple Support document I posted is applicable. Same goes for any inability to download documents. If you click a download link the standard macOS "file save" dialog should appear. Describe what happens instead.

"Running applications" requires additional explanation.