Is there an error code or number? What video formats?
Just YouTube or are other video websites affected?
Try another iPad if possible. Determine if the problem is unique to your iPad or maybe stays with YouTube and/or other video websites.
Any recent iOS or app updates on your iPad?
Check Apple's website - there may be some information to be found.
Also check YouTube's website. Could be something there.
The objective is to narrow down the problem's source or reason.