I've had this issue too. Seems the phone fights over using wifi or packet data as you say. It does auto switch. I assume it sees a better 3G than the wifi so wants to use it. I've also found power saving mode interferes because the phone wants to turn off the wifi. It might help to check your settings for power saving mode. I un-checked the auto off wifi setting.
As to browsers, I've started using Opera Mini, despite the comments on PlayStore. I find it excellent. Especially having an easily accessible refresh button when using forums