Within Teamwork.com, you can choose to show only the items you have been notified of in the notification bell area.
Click your profile image in Teamwork.com's main navigation menu and select Edit My Details from the pop-out menu.
Go to the Preferences tab and toggle on Show only notifications for items I'm following.
Click Update to save your changes.
Going forward, you will only see a badge count for new notifications next to the Notifications bell when you have specifically been notified of something.
For more information, see: Reducing Project Email Notifications