The default YouTube app on iPhone hasn't received an update since it was originally released. In fact, the new, mobile version of YouTube is more feature-rich and faster than its app counterpart. If you're sick of your iPhone always opening up the app instead of the mobile page, Mac OS X Hints has a simple way to stop it from happening.
Since you can't delete the stock apps from iOS, you have to block them from opening. This is done by setting up a restriction. Tap Settings > General > Restrictions. Flip the YouTube slider to off. Now the YouTube icon will disappear and any time you want to load up a video it will will stay in your browser, no app required.
Force YouTube Links to open in mobile website on iOS [Mac OS X Hints]