I am not a lawyer. I have talked to lawyers about using the GPL and LGPL for code to be interpreted / dynamically linked into non-free software. We all got big headaches. The question isn't just whether you have to release the rest of your site under GPL. It's whether a non-free browser can legally run GPL code.
My best non-legal advice is to never try to use the GPL on Javascript code. I'd never release a Javascript library under GPL, and if I found some code I wanted to use I'd try to get the author to give me an exception. The LGPL may be a better choice, but that license is so complex I suggest avoiding it as well.
The FSF themselves have an answer for you in their FAQ If a programming language interpreter has a license that is incompatible with the GPL, can I run GPL-covered programs on it?. That answer itself is frustratingly ambiguous, but it suggests paths where you could run GPL code in a non-GPL browser. You may also find the FSF essay The Javascript Trap interesting, although again it doesn't really give a clear answer to your question.
For context, it helps to remember that the GPL and LPGL were written in the Unix era when most everything was statically linked. No dynamic linking, no interpreted languages where the scripts themselves were considered valuable intellectual property, etc.