After clicking on my username I can click on 'Your Alerts' - and I can see a list of alerts I have had - they don't bother me because I can't normally see them, why do they bother you? You perhaps feel that they are like 'read' e-mails that you no longer need, is that it?
Just don't click on your username and 'latest alerts', - problem solved !
I am guessing the list of alerts won't continue to grow, it will just operate on a FIFO principle with the oldest being automatically deleted. In any case this is something for the site web admins to think about, the alerts will accumulate on the server and use up disk space.