Giving a teenager a cell phone with Internet is very dangerous. It is like a loaded gun. They can access the Internet any time from any where without any supervision to view pornography, order drugs, order weapons, view scenes of explicit real-life violence and be targeted by predators. As a parent you will be out of the loop. Your child has the potential to be out of control.
Most cell phone providers will let you block Internet on your children’s phones even if they are Internet capable, but if the phone is “wi-fi” capable like the iPhone, they’ll access the Internet anyway from public Internet connections. The best solution is to give them a phone that is not Internet capable. Sell their current phone on ebay if you have to.
If you chose to allow them to access the Internet from the cell phone, at least check the history. Many phones, like the iPhone, will keep the history and only allow you to delete all of the history, not selected entries. Make an entry in their Internet history and make sure it is still there when you check the history. If the history gets too long, delete it all and make another reference history entry. If the reference entry disappears, so should your child’s cell phone.