I have done a quick scan, it didn't change anything. I have done a full scan, it didn't change anything. I rebooted, same thing. I cleared the temporary files of the browsers, again, same thing. I also sent an email of support to the Twitter team and I am awaiting an answer on this.

As of today, IE and Firefox are completely messed up on Twitter, either logged or not. With Chrome, I am able to use the site.