You can also use `H` for back and `o` for opening the URL prompt, among others. That requires you to be in the insert mode for typing in input boxes or interacting with the website you're on via keyboard shortcuts though.
Since you mentioned Bitbucket's file search, fzf[1] immediately springs to my mind. I don't know how well it would scale to your use case but it supports pattern and regexp based searching.
I'm guessing that it has to do with the focus behaviour of the mouse. If that's the case then it would be the 'focus_follows_mouse' variable. It is set thusly:
https://github.com/ogham/dog is pretty good in that regard