That strikes me as odd enough I must ask: what model phone do you have, and do you have a cell-data plan enabled for it (in addition to, or as part of, your regular phone/text plan)? Or to put it another way, are you able to use the phone to browse the internet with Safari while the WiFi is turned OFF?
I have a 4s model. I have WiFi enabled and do search the internet with cellular without wifi. The only thing I can think is I don’t stream videos with the phone and have no add-on apps to do such.