You're not really giving us enough information on the network connection. I'm assuming it's connected via wifi due to the wifi symbol, and that you're getting a pretty strong signal since the wifi symbol is pretty much at max.
What's your ISP and what router are you using? What's the speed like on a PC connected via cable?
And you said you tested the iPhone 5 on a different network... does that mean you actually tested your S4 on the mobile network? What network are you on, and what network is the iP5 on?