Commercial Fixed Wing
Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Air Transport Pilot (ATP)
Situational Awareness
Conflict Airborne Conflict; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Clearance
Flight Crew Returned To Clearance
Environment - Non Weather Related; Human Factors; Procedure; Aircraft
Climbing to 7;000 feet with Aircraft Y traffic at 8;000 feet MSL.  Turning from departure heading of 035 to assigned heading of 320.  Received a TCAS TA and a TCAS RA at about 6;700 feet MSL to 'Level Off.'  Complied with the RA and notified ATC.  Traffic passed above and behind us.Climbing turn near traffic with light aircraft climbing faster than normal (235;800 lbs).Shallow climb rate near traffic.  We were very light with a takeoff gross weight of 235;800 lbs.
We were climbing to our initial level off altitude of 7;000 feet on a 035 heading. Departure alerted us to traffic at our 12:00 O'clock at 8;000 feet and had us turn to 320. At approximately 6;700 feet; the TCAS alerted us to traffic and then in the turn leveling at 7;000 feet the TCAS alerted us with an RA to level off. We complied and traffic passed behind us. We alerted ATC of the RA and are filing this report.Light aircraft (235.8 lbs) out of ZZZ maybe confused departure and we climbed to altitude faster than other airline aircraft usually do. Also busy terminal area with lots of GA traffic all could be potential causes. No one felt threatened likely in either aircraft but RA did go off.Earlier vector from departure away from oncoming traffic despite altitude separation.
Air Carrier flight crew reported over shooting an assigned altitude with a light aircraft; resulting in an airborne conflict.