My iPhone 5c will not turn on

I took out my SIM card and tried to put it back in again because it was saying that I did not have one in there, even though I did, and the SIM card accidentally got stuck in the slot, without the insert tray… Not really thinking the whole thing through, I tried to get it out using a paper clip etc… But it was stuck. So I tried to pry open the screen of my phone using some tweezers and I got it open slightly enough so that i could see the inside of the phone and the battery, despite the two bottom screws still being inside. Once I did this and got the SIM card out and then pressed the screen back in, my phone screen went black but the light inside was still on. I tried turning the phone off after i propped the screen back in and now it will not turn back on! I think I may have broke it or something? Does anyone know how i could try to fix this myself, or if I'm gonna have to take it to the Apple store… Could i have maybe unplugged one of the wires to the battery when i pried it open?

What you did either dislodged the ribbon cable to the screen or damaged it. The ribbon cable on all iPhone 5 models is very intolerant of any kind of abuse or poor fitting.

The fix is to *correctly* and carefully remove the screen, check the ribbon cable for cracks, and to then refit it. Cracks in the conductor tracks on the ribbon cable are not always visible though, and just a single crack on a single conductor track is all it needs stop working.

The 5 and 5c are about to become very cheap as used phones following Apple's announcement earlier this week that they will stop all support for those models when they release iOS 11 soon. So if you have to get a professional repair then the cost might not be worthwhile. New screens are cheap on eBay and often come with fitting tools.

The trick when fitting and removing the screen on the 5, 5c and 5s (also the SE), is to not lift the screen further than 90° from the horizontal phone body as that overstrains the ribbon cable. See YouTube.