Copying the reply from Please Explain "CSRF Failed" over to here for anyone else who is interested in this:
So, what the private browsing mode test does is to quickly check to see if it is likely an add-on / extension that you have installed that is causing the problem, since when you’re in private browsing mode most addons / extensions are disabled.
To figure out which is causing your problem, you can walk down the list and disable them in turn until it works, then figure out if you want to just live with the pain or uninstall the add-on 