Restarts after screen replacement

There are a few things that could be causing this issue.

1. Re-install the original screen. If it still works, even though it's cracked, it can still serve it's purpose for troubleshooting. If the original screen works fine, then the replacement is either defective or something went wrong during the installation.
2. Closely inspect the flex cables on the replacement screen and the connectors on the logic board. Look for signs of damage or debris that would impede a good connection.
3. Did you keep track of the screws holding the screen retaining bracket? This is critical, especially on the iPhone 5S as all the screws are different length. If you put the wrong one in the wrong hole, you can damage the logic board in what's known as Long-Screw Damage. This would require a micro-soldering repair and is the worst case scenario for your phone.