Light jet Captain reported the Garmin 5000 failed to execute the charted turn at the required rate during initial climb on the Mecca1 SID departing TRM.

Narrative

Prior to departing runway 35 (TRM); the Mecca 1 Departure was loaded into the Garmin 5000. NAV mode was selected. The departure specifically calls for a departing right turn to a heading of 195 degrees; followed by a course interception. Upon departing; I noted that the NAV course had us beginning a right turn. I was hand flying initially; and followed the initial course turn to the right; as prescribed by the Flight Director. Autopilot was engaged sometime after FRA. I noticed that the automated right turn appeared to be happening too slowly. I paused briefly; thinking that the turn rate would increase; but I became concerned when the rate continued at what I would describe as a; half bank;" rate. I expanded the MFD map preview to see that the Garmin had drawn the right turn as a gradual; shallow bank turn to the right; at a rate that was not keeping us the proper departure profile. I immediately deselected NAV and selected HDG; followed by dialing the heading bug to the proper heading of 195 degrees. As the aircraft began to turn right at the proper rate; ATC queried us as to our flight path. The ATC controller sounded annoyed as he asked us if we were flying the departure. PIC acknowledged that we were flying the departure. ATC responded with several comments; including that we were not on the departure and that he was receiving a low altitude alert. He then commanded us to make a LEFT turn to a heading 180 degrees. PIC acknowledged. I turned the heading selector back to the left and began the left turn. Seconds later the controller again contacted us exclaiming; "what are you doing?!!" I stopped the turn as there was now an obvious element of confusion created. The controller continued to inform us that we were not on the departure; that we were at an incorrect heading and that we needed to climb to 8000 feet immediately. PIC again acknowledged and we began a climb to 8000 ft. The controller was clearly agitated; and turned us to the right; directing us to a fix. He continued to scold us; for lack of a better term; and questioned what we were doing. I then responded to him and stated that after departure; our flight director commanded a gradual turn. We recognized that discrepancy and began to increase our turn rate to the right; but at that same time he had given us a command to turn left to a 180 degree heading. I stated that we were complying with the turn that he had given us. His response was that he was curious as to what had happened; and he did not question the situation any further. A frequency change was then given to us; followed by a pleasantry of; "Have a nice day." To be certain; clear sky's prevailed with unlimited visibility. We were fully aware of where all terrain was located in our vicinity; and never positioned ourselves to be at risk of proximity with the terrain. I am not submitting this report because of any misunderstandings with ATC. I am submitting this report because of the failure of the Garmin 5000 to command a proper turn during a published departure. I have experienced this same issue previously while flying with the Garmin 3000 avionics. In previous instances the Garmin would command a half bank turn while on departures or arrivals; as well as in a holding pattern. While I am relatively new to the fleet; this is the first time I have experienced this with the Garmin 5000. It should be noted that current company guidance exists; warning that the Garmin will sometimes command an improper turn while departing TEB runways 1 and 6; specifically while executing the Teterboro Four Departure.As for my perspective of this incident; I fully acknowledge that I departed from the prescribed route of the Mecca 1 Departure. It was not for a lack of understanding of the departure itself; nor from a lapse in situational awareness. I flew the precise course that the Garmin 5000 was commanding; and recognized that it was not guiding a proper course turn. I inappropriately delayed my response too long; as Iwas believing that the Garmin would increase the rate of turn at a certain distance from the airport; or at a specific point during the initial turn. By the time it became obvious that the Garmin course was not correcting itself; I had allowed the aircraft to exit the proper departure course. As I increased the turn rate to correct the issue; ATC intervened and turned us to the left; completely removing us from any possibility of reacquiring the departure course. Any indication that the Garmin is lagging in a turn or displaying an irregular course line must be immediately intervened with a manual turn our other corrective action. Having the prior knowledge that this type of error can occur; is a key factor in monitoring and reacting to this type of situation; and responding in a timely manner. In my case; I allowed the error to progress too far before intervening.I was lulled into believing that the Garmin automation would self correct. I will start the practice expanding the view of the MFD after loading a departure; to review any potential turns that the Garmin automation has drawn that appear improper. It is common for most of us to focus on a view of 2.5 miles to keep a proper situational awareness of other traffic during the departure. Expanding that view; at least temporarily to confirm proper course turns; might expose a potential problem with an impending improper turn situation. This incident was fortunate in the sense that it did not interfere with other traffic. I believe that information should be made available to crews that this type of error can occur with the Garmin avionics suite. I know that some crew members are already aware of it; while others may be learning of it for the first time."