How can I stream tv shows on my iPhone 5s (without downloading)? I've looked everywhere for an answer to my question but I can't seem to find one:

I'm trying to watch a tv show on my iPhone 5s that I bought on my account via the computer. However, the only way it lets me watch them by downloading the episodes, and I want to just stream them on my iPhone because downloads use too much data. How do I fix this problem? Is it possible to stream tv shows on your iPhone?
I'm sure there is an answer for stuff downloaded from iTunes, but since you didn't specify, then the standard solution you are looking for is PLEX MEDIA SERVER -- https://plex.tv.

They offer a Plex Pass, which you have to pay for in order to receive extra features, but I believe the free version has everything you need.

Setup the media server on your home computer.  You can even setup a NAT port on your router to access it externally, and it can either support authentication via Plex Pass or there is a hackish script somewhere on Github that allows authentication without having to purchase a Plex Pass subscription.

Then download the app on your smart phone, and you will be able to stream all of your downloaded movies that you are sharing from your Plex Media Server.  I do this all the time, and it works perfectly.  It supports transcoding on-the-fly, so you never have to worry about buffering and such.

XBMC, which is called something else now, is also an alternative, as well as OpenElec.  I've done my research though, and Plex is the definitely the most full-featured of them all.

The setup is even simpler if you only plan to watch on your own wireless network and not over the Internet.  Your mobile app (or smart TV) can find the server on its own, and it's pretty much set it and forget it.

There are multiple apps that you can download for Plex as well that integrate with Couch Potato, Sickbeard, etc.  I do this all the time for my kids in the morning when I'm making their breakfast and they want to watch something that's not OnDemand or DVR.  I just have all of their favorite shows downloaded to a folder on a Mac Mini running Plex Media Server.

I fire up the app on the iPad and get a few minutes of morning peace and quiet while getting breakfast ready.  It's pretty much perfect in all aspects.

Also, to answer your question in regards to using data, if you stream over your wireless network, you will not use any additional mobile data, as long as your mobile device is connected to WiFi.

I know your question is regarding watching on your mobile device, but here's some additional helpful info for other users:

It should be known, that not all smart TV apps offer full integration with the free version of Plex.  For instance, my parents Samsung smart TV has full Plex app integration, no problem (as does Google Chromecast).  However, my Vizio smart TV requires a Plex Pass subscription in order for the app to even work at all.