Loading Problems in Safari 13.1

Hi all,

I'm having an issue in Safari where on certain webpages, particularly on a password protected one, embedded images and videos won't load. Where the image would normally be is a square with either a question mark or the name of the image file. Videos don't load either, and instead show a message saying I've blocked third-party session cookies. This happens even though I have the "Block all cookies" option off in Safari privacy preferences. This issue only started happening recently, possibly since updating to Safari 13.1 (but I can't remember exactly). This doesn't happen in other browsers (e.g. Chrome or Firefox).

I've read other pages on this forum where users seem to have a similar issue, but they're using either High Sierra or Mojave, then reinstalling macOS in safe mode. I'm running macOS 10.15.4, the latest version of Catalina, and so these fixes don't seem like an option for me.

Any suggestions?