Not OP, and YMMV, but i never had problems with google, amazon or any other website. ClearURLs is one of those extensions you install, after you open your browser for the first time and completely forget about it, because it just works (TM). As a matter of fact, I had to make sure that it was even installed, before commenting.
For what it's worth I use the FF version and I've never had this problem as far as I can tell.
I also use the FF version and have not encountered this problem (to my knowledge).
It does break youtrack quite severely and it took me forever to single out this extension. Afaict there's also no way to filter URLs so I had to get rid of it on my work browser :/
I don't use Google OAuth much but it works for when I need to enter YTTV credentials. It has never broken Amazon checkout for me, smells like FUD.
I used to use this and it did indeed break Amazon checkout. Took me forever to figure out!