I replaced an iPhone 5c screen with a fairly cheap replacement from Amazon. After putting it all back together and turning it on, the screen was unresponsive/laggy and wasn't registering touches accurately. I opened it back up, disconnected the battery and screen connectors, wiped the back of the front screen with some isopropyl alcohol, and cleaned the connectors with isopropyl alcohol. After reconnecting everything, the issue persisted. I again disconnected the battery and screen and then reconnected them. This time it worked perfectly! I guess I just didn't have the connections properly seated the earlier times.
My suggestion is to clean the connections and keep trying to make sure you have the connections seated properly. Good luck!