This seems to be a pretty widespread problem and no "formal" fix has been implemented. It's ses you have taken all the preliminary steps with your network to be sure that it is not the prob. I'm on my 6th iPhone (5) and what I see as the problem is 2 fold 1) competion within the hardware as to obtaining wifi vs. 4g and 2) a possible conflict as to requesting a wifi network. These solutions may not be foolproof as it appears that hardware/software fix is most likely necessary but I hope one of these works for you:
1) settings>cellular>turn "off" cellular data. (Note: you can still get text's bit will not be able to get picture attachments).
2) settings>wifi>ask to join other networks turn "off."
Good luck !!! Please post back how it went.