When you go to the YouTube site in Safari, scroll to the bottom of the page. You'll see a link to Desktop. Tap it an you will be taken to the full desktop version of the site.
Alternately, and for web sites that don't offer Desktop version links, you can use the Atomic web browser. Once you load this app, go to the little gear icon, chose Identify Browser As, and choose what browser you want the website to think you are using. If you choose one of the non-mobile browsers, all web sites should show up in their full webtastic glory.
That's not to say that the iPad will necessarily work well with the site. There is generally a reason why those sites provide mobile versions, and it isn't always just about the screen size.