While showing my work to my supervisor, an adult website appeared. Am I in trouble?
It's embarrassing, but it's also understandable. I think if your supervisor has decided to ignore it, then you should take this opportunity to not ruminate. So silently thank him, and forget it.

I know advice is always easier said than done though. Rumination is defined as "to keep thinking about a problem which had already been, or can never be solved". The real problem had already been solved at the moment he decided to ignore it, so the only problem here is your rumination. To stop it, every time you feel embarrassed about this topic again, my tip is to shake the idea in your head, like it's just a toy. This will make you distract from the negative feeling. 



Related:
• Why You Should Stop Caring What Other People Think (Taming the Mammoth) – Wait But Why
• Straightforwardness, Fearlessness and Improvisation: How to find the fresh perspective? – Ooker