New to you tube

YouTube TV has an unlimited DVR, so there is no way to delete individual episodes from your Library. It'll keep track of what you watch, so it's easy to tell which episode is next in your queue to watch. 

 

(Note that if you don't see "WATCHED" on the episode thumbnail, be sure your watch history isn't paused under Settings > Privacy).

 

You can remove a show from your library, by clicking the check mark under the show's title, but this removes the entirety of the show, not individual episodes. If this is what you want to do though, go to the show page and click the check mark at the top.