The most logical explanation is that you have a poor quality replacement screen. Where did you buy it?
If it came from a reputable supplier like iFixit, then I would contact their customer service and ask for it to be replaced under warranty. If you bought it through eBay, you also have some recourse but know that the most, if not all, of the worst quality screens on the market get sold through eBay. That doesn't mean there aren't any good ones, just that you have a higher likelihood of getting a bad one. Also, some of the earlier and cheaper 5S screens don't work well on the SE.
There is of course a possibility that your logic board is causing the issue. I would try to re-install the original, cracked screen to see if the problem goes away or not. You can find a good dis-assembly guide here.