Hi Caran,
I found a video with a solution (4:47 but rambles a lot and can be summed up very few words, see below).
http://youtu.be/0mQPsyObIz8
Slightly ironic, given video playback is the problem but at least you should be able to view it in Safari or IE
The person who made this video believes that the problems emerged after an upgrade to IOS6. I'm not sure if this applies in your case?
Anyway, from the Home Screen:
1) Go to Settings
2) Scroll down and click on 'Reset'
3) Select 'Reset all Settings'
4) A popup warning appears but assures you that user data and programs will not be lost
5) Click Continue.
When you return to YouTube, its appearance will have changed and the problem videos should now work.