The website says that it takes about 2 weeks, and yet my run hasn't been submitted on the forum after I requested it on June 13.

Do you mean you requested a new game to the site, or that you submitted a run to a certain game and it wasn't yet verified?

If it's the first, due to the current circumstances the game requests queue is slow right now, and it takes site staff more time to accept games. Just be patient.

If it's the second, try to contact the moderators of the game in question (although by site rules they are allowed 3 weeks to check runs).