Apple - How can I make You Tube full screen on my iPad when using Safari?

Use the 'desktop' mode instead: it's found at the end of the YouTube menu (click on the logo at the upper left).

The fullscreen YouTube uses of its website, is not a "real fullscreen" feature. It just maximize the size of the video to the size of the webcontent.

So unlike Vimeo (for example) you won't get a real fullscreen without YouTube updating it's webplayer (or you using the YouTube app) .

I had the same problem until I started watching the videos through Google videos instead of browsing through youtube. That way even if the video you want to watch is on YouTube you can still make it full screen through Google videos.