Chosen Solution
I’m currently having issues with a new replacement LCD for a IPhone 5s that I planned to install for a friend in her phone. However, this LCD required me to transfer all the additional part (camera, earpiece, etc.) to the new LCD. Upon reconnecting the all the cables, and before closing the device up, I turned on the phone to test it, and noticed that when attempting to fold the flex cable into place the screen would fade to white. But when returning the screen to a 90 degree angle to screen would function as normal. In addition, the new screen had no touch capability whatsoever. I noticed a small piece of plastic sticker pealed off the digitizer, revealing three contact points.
Ideally, if you can easily replace the screen under warranty, that should be your first option. However, if that is difficult or impractical, cut a small piece of electrical tape to cover those exposed contacts. When you fold everything in place, they touch the LCD backplate and short out, which is causing your problem.
Try a new screen and return the old one, sounds like a crappy screen.