Hello Alison9999,
Welcome to Apple Support Communities. We understand you have some issues with opening certain websites on your Mac and that you tested the issue in safe mode.
First, we'd recommend clearing your browsing history in Safari to see if that helps. You'll want to restart your Mac after doing this before testing those websites:
Clear your browsing history in Safari on Mac - Apple Support
Next, the goal will be to rule out what's going on. Test in another web browser application. Also see if Private Browsing in Safari works. Finally, create a test user profile, log in to it, then see if those websites open. This will help rule out if the issue is related to your user profile specifically.
Set up users, guests, and groups on Mac - Apple Support
Keep us updated with how it goes.
Best regards.