Although these two words don't on the surface look very similar they do sound linked.
Bezig is pronounced /ˈbeː.zɪx/ and busy is pronounced: /ˈbɪzi/
So not exaclty the same but happily similar to help in remembering the link between them.
You won't be surprised to learn that they have the same root and that busy is "cognate with Old Dutch bezich, and Lower German besig." Reference.
I'd be happy to hear some of the subtle differences between the words 'busy', 'bezig' and 'druk'...