I don't have much to offer, but here's some general information.
Are you watching the YouTube videos in the YouTube app, in Safari, or in another app (in app browsers).
As it should be, the YouTube app is usually the most reliable way to watch. Restart the iPad if you having problems. This will fix it, most of the time. Provided the internet connection itself is not the issue.
Safari should work, but if you have an older iPad or the internet connection is touchy, it can cause problems. It's also possible for Safari to have other problems that show up more noticeably with videos. It may be necessary to restart the iPad or clear data and history from Safari to get the best performance.
Watching YouTube videos in third party apps like Twitter, FaceBook, or a news reader varies a lot. It really depends on the app, the particular video, and Murphy.