Light twin engine pilot reported an NMAC with a light aircraft in the vicinity of LEE airport.

Narrative

After being cleared to enter a left downwind for Runway 4 by the Controller. A departing aircraft made a left hand turn approximately 60 degrees left putting them on a conflicting path. At the time we hear the Tower asking the pilot their heading and to perhaps stay on runway heading. It never occurred to me as the pilot that they could have departed and already be at or over 1500 ft. AGL (we were still descending to the pattern altitude). We were also busy looking for two aircraft that were ahead of us in the pattern. The pilot had been having trouble hearing the Tower and told them they were fixing the radio. Then Tower told us to look for traffic. The traffic alert on my iPad then alerted me to the traffic. I looked up and immediately turned right and dove watching the aircraft pass over our left side; less than 200 ft. from us. Going back and listening to the ATC recordings the pilot had been cleared for a left turn on course after takeoff by Tower but looking at the ADS-B track and ours it appears that they may have turned early (the track does not appear till after the near miss). In hindsight it was a mistake to not be already at pattern altitude that close to turning downwind; and also to dismiss the departing traffic and only be scanning for the traffic in the pattern we would need to slow for.