There may be some trailing log out / sign off chatter (Quora web server - Brave Browser) wherein, what Quora wants, Quora does not get.
And that might be involved with, or not, the path you took, when you signed on.
Sometimes a web server sends some trailing pop-up window that has a message confirming your log out . . . and maybe, the browser displays that, or not.
You might supply, here, screenshots of that, but at least, more specific detail re what you did for “Logout” and what occurred, that you “observed immediately” ?
Something alone these lines seems to maybe have been the solution. I was unable to find the repeatable steps for resolving the problem. I am not even certain cleaning browser and restart device had anything to do with solving the problem, because the problem did not disappear immediately after trying this path. I don’t know what constitutes “cleaning browser”. I deleted cookies and verified no data was in storage for this site, restarted the phone. When I then tried surfing to Quora, I got an error message from Quora saying I needed to enable cookies for the site. In site settings for that site, cookies were enabled. A few days later, I went back to Quora and it worked as with no sign of a problem.