Air carrier crew reported an air conflict with a single engine aircraft just after lift off. The single engine Instructor Pilot; who also reported; was landing on a crossing runway stated the Tower had given them clearance for the option therefore; decided to make a 'touch and go.' After receiving an alert to turn crosswind from ATC; the Instructor saw the Air Carrier and took immediate action to avoid a collision.
Synopsis
Air carrier crew reported an air conflict with a single engine aircraft just after lift off. The single engine Instructor Pilot; who also reported; was landing on a crossing runway stated the Tower had given them clearance for the option therefore; decided to make a 'touch and go.' After receiving an alert to turn crosswind from ATC; the Instructor saw the Air Carrier and took immediate action to avoid a collision.
Narrative
During a normal takeoff and climb from Runway XX at the ZZZ; we saw a single engine aircraft at our altitude turning away from us. At approximately the same time we heard the Tower Controller tell the single engine aircraft to turn crosswind now. The single engine aircraft was at our altitude and very near to us. We did not have any time to react as we did not see the single engine aircraft until it was wing high turning away from us. Apparently; the single engine aircraft was supposed to do a full stop landing on Runway XYR and either did a touch and go; or a go-around from that Runway. I only first saw the single engine aircraft about when I was retracting the landing gear; and that's when I also heard the First Officer comment on the aircraft.The horizontal distance was likely much less than 1000 feet; possibly even 500 feet. Had the single engine aircraft not reacted; a collision could have occurred at the Runway XX intersection with the Runway XYR extended center line. The single engine aircraft turned away between us departing XX and the Control Tower; for further geographic reference. Tower frequencies may have been split at that time as I heard the Tower Controller; but not the single engine aircraft on frequency. It's also possible that I just may have missed the single engine aircraft transmission. From our vantage point on Runway XX; our visibility of Runway XYR is not good as structures and aircraft block it from our view. Controller staffing limitations may have been a factor here too. Captain Response; Controller staffing needs improvement.
Second reporter narrative
We were 'cleared for takeoff Runway XX full length; shortened'. I was the Pilot Flying. During our initial climb at approximately 300 ft. AGL; I heard the Tower Controller tell an aircraft 'turn crosswind now; turn crosswind now; you were cleared for a full-stop only'. I looked outside my forward side window and saw a single engine aircraft in a right bank at our altitude; approximately 500 feet from our aircraft. The aircraft was operating on the crossing Runway; XYR. I did not maneuver because by the time I saw the single engine aircraft it was already clear of our flight path. After seeing we were clear of the single engine aircraft I looked back inside and saw a yellow TCAS TA icon on the Navigation Display. The Tower Controller told us to switch to Departure and apologized for the incident. Of note; the first half of Runway XYR is not visible from the approach end of Runway XX due to large buildings between the two Runways. First Officer response; maybe a more diligent outside scan when flying in and out of airports with a lot of student Pilot Training.
Source: NASA Aviation Safety Reporting System (public domain). Reports are voluntary submissions and are not verified by NASA.