Brave
Ok I managed to solve this issue with the help of @AnInternetTroll. This is a heads up for other users that might also encounter this issue. For short, what happened to me was that there was a PHP setcookie on Brave that wasn't getting overwritten whenever I tried to log in. When you log in on Speedrun.com, if your password is correct then you're going to get a PHPSESSID which is basically a cookie that the browser sends to SR.C to identify who you are, and it changes every time you log in. What happened to me on Brave however was that there was a PHPSESSID already there that wouldn't get overwritten when I logged in on Brave, and instead I would get a second one on top of that one which would immediately disappear and thus I wouldn't get logged into the site. The reason why it worked on Chrome is that there wasn't a cookie already there, so when I tried to log in with Chrome it worked.
Now, how do you solve this issue? Press either Ctrl+Shift+I or F12 to open up the Console, then click on the >> next to Network or Sources, click on Application, then go to Cookies and click on https://www.speedrun.com.
What you want to do next is click on Filter and search "php".
If you are already logged in, there should be a PHPSESSID (don't delete it if you're logged in already), but if you aren't logged in and there is a PHPSESSID then just click on it and press Delete.
After that you should be able to log into your account again.
I use Brave and something kinda weird happened to me about a week ago. I accidentally logged out of my account and because of 2FA I had to get a verification code to log back in. But while I was going to my email I hit log in again and it just let me. I didn't even open up the email with the code yet.
Thanks for the report, definitely a bug - we'll work on a fix!
