I hate it, too...but...
Always when I try to play a YouTube video on my iPad it automatically switches to the YouTube app. I don't want that!
...but, do you actually mean using the Youtube app, or Youtube "mobile"? I haven't tried the Youtube app, but I'm sure it sucks. Like you, I just want to use the normal, full Youtube site. When I first got my iPad 4, and went to Youtube, through Safari, I was automatically directed to the mobile version of Youtube -which also sucks. I figured out how to just get to the regular version, though:
If you are in the "mobile" version, you can find a button that will allow you to go to the full site. You will have to poke around, a little, or, just use this link:
http://www.youtube.com/?nomobile=1
Note the "nomobile" part.
For ease, and because you will automatically be taken to the mobile version otherwise, bookmark the link.
Hope this is helpful and best wishes.