If YouTube app is nowhere to be found on the phone then try checking if it's not restricted in settings> general> restrictions/
If you can see YouTube in that restriction list then your iPhone is one with YouTube censorship.
IOS6 which is the current operating system of the iphone5 and any apple device owner who chooses to install the update, will not have the you tube app. because apple and youtube did not get along and ended up with ios6 not having the app. the only way to access youtube is through the app youtube launched called "youtube" its not the best app but we'll have to live with it for now. kudos