I had this problem on my android phone visiting chess.com in the browser. I've clicked on single messages but the counter wouldn't go down. Solution was to scroll down to the very bottom of the alerts menu and click on "clear all".
Hope that works for you, too.
If you're using the android app, try following: First off, delete the app cache and the app data via your android app storage settings. Then remove the app completely and reinstall it.
If not, contact tech support, they're helpful and answering quite quickly.
Can't Delete Alerts In Android Mode !
Up to 500 so far !!!