How long does it take for something to not show up in search, if ever

So when I was younger I made an account to a site. Stupidly, I used a common username that can be associated with me. It’s not world ending but just something I’d rather not people find haha.

I changed the username a few weeks ago but the old name still shows up when searched for. Does anyone have any idea how long it’ll take for that to stop. Or is it something that never goes away because it’s archived on the internet or something like that? Thanks!