iPhone Safari will not allow me to log out of YouTube

Hello, my friend used my iPhone to log into his YouTube account on Safari over a year ago. He did not save his password or username in Keychain--I've searched it and cannot his log in details. Yet, for some reason, I am permanently logged into his YouTube account in Safari. Even when I tap on his icon and tap log out, it won't log out. The page will refresh after I tap log out, but I am still logged into his account after the refresh. I can repeat this ad naseum, but no matter what, I am still logged into his account. I can't add my account either. I also can't delete his entire YouTube account because I do not know his password. (We aren't friends anymore, so it's not as if I can ask him for help.)

I have no idea how to log out, when the log out button won't work. I don't want to download the YouTube app because that doesn't solve the problem. It merely redirects any Safari search for a video on YouTube to the app, without ever resolving the issue of me being permanently logged into his account. I want out of his account for good before I download the YT app.

As a matter of personal security, it does strike me as odd that someone can inadvertently be logged into another person's device permanently.

The model iPhone doesn't matter because I've already upgraded my phone in the meanwhile and the problem persists. I had an iPhone 5c, now I have an SE.

Also, I tried asking YouTube but they told me to ask Apple. So no help there. Please help me Apple Community!