In my short time here, I already got a lot of Discourse notifications:
According to this meta.discourse post,
Notifications go away when you interact with the cause of the notification, either by clicking the notification or navigating manually to the cause […]
However, my notifications don’t go away like that, and in addition to that, the Dismiss All button is disabled on my page https://discourse.slicer.org/u/jondo/notifications.
Does this work for you?
I don’t think there is any way to clear the list. Viewing a thread or clicking “Dismiss All” will get rid of the unread count above your user icon (number inside a blue circle, e.g.: 
) and the light blue highlighting of each unread row in the activity list.
Related: How to delete/clear notifications? - #6 by jomaxro - support - Discourse Meta
Ah, I expected the items in the small popup list to be removed, like the Discourse founder had implemented on StackOverflow. I have told the Discourse people now.