A CE560 flight crew failed to comply with the track requirements of the Teterboro SID from TEB.

Narrative

While departing via the Teterboro Six departure for Runway 24; situational awareness was lost by both flight crew members. I was the pilot flying and prior to engine start had reviewed the departure procedure; which calls for an initial climb heading 240 to 1;500 FT; then a right turn to heading 280; crossing 4.5 DME TEB at 1500 FT; then maintain 2;000 FT. During taxi; this procedure was also discussed as part of the departure briefing. Immediately after departure; we encountered continuous moderate turbulence. I climbed via the 240 heading to 1;500 FT and leveled off; however I did not execute the right turn to a 280 heading. For some reason; I had decided that we were not to turn until crossing the 4.5 DME of TEB VOR.The Captain was switched to Departure Control prior to our level off; and reported our climb to 1;500 FT. After leveling off; and continuing on the 240 heading; a very irate Controller asked if we were 'turning right; like you're supposed to?' At this point the pilot not flying advised that we were; and I immediately began a right turn to a 280 heading. The Controller also advised us of a possible traffic conflict with a Boeing 757 on approach to Newark at our 11 O'clock; and 3/4 of a mile who would shortly be descending through our altitude if we didn't hasten the turn.Neither the Captain nor I had recognized the mistake; until it was brought to our attention by the Controller. I was busy trying to keep the aircraft under control and maintain the altitude; and I believe the pilot not flying was too concerned with my struggles with the aircraft to recognize the problem as well. Had the Controller not said anything; I believe we would have continued on the 240 heading until 4.5 DME; before executing the rest of the Departure Procedure.Given the poor weather; and the congested nature of the airspace; I believe it would have been prudent to give up my leg of the flight to the more experienced pilot on the aircraft.