I have the same issue. I have a 2+ year old IPhone 6 with software update 10.3.3 and over the past 3 weeks the touch screen is not responsive immediately. Sometimes the lag is only a few minutes but sometime it is several hrs. After the lag, the phone tries to execute all the touches at once causing chaos on my phone. Apps when opened seem to work fine and Siri and the home button function fine. However the problem continues and appears to be a software issue.
Steps I have taken already
1. Hard restart of the phone. ( I do this often now to end phone calls when the screen refuses to work)
2. Reset all network settings
3. Reset all settings
4. Brought to Apple Store and restored the phone from recent backup. PHONE worked fine after that for about an hour but then same problems reappeared.
5. Want to get some better advise from Apple. Apple employees advised to reset the phone as a new iPhone (which would mean I lose everything I had from my last backup. (Messages, prior phone calls etc).
6. Is Apple able to stand by their product updates on the day they launch the newest iPhone ?