I use PyGTK quite a lot, and my code is hosted on github. I know that there are many websites that can run Python code, but I was hoping that maybe there is a website that can handle the new windows created in my code in the browser, so that people could test my code without needing to download it. Is there such a website?
No, there is not.
If you switch to Python 3 and from gi.repository import Gtk, then you can use Broadway to run a GTK application in your browser. However, this only works locally, not as a website.