Helicopter pilot reported an NMAC during departure with a military fighter doing a high speed pass over the runway. Tower controller issued helicopter a takeoff clearance which put them into the flight path of the fighter.

Narrative

Aircraft Y were arriving at ZZZ for an air show in the following days. Each Aircraft Y does a high speed fly over and break before coming into land. I had just refueled at the FBO after completing a flight and had a follow on flight to ZZZ so I was medevac status. The Aircraft were landing on Runway XX. I informed the Tower that we were medevac status with a destination of ZZZ and stated my on course heading which I believe was 230 degrees. I requested a departure from the intersection on X and Y Taxiways with the initial takeoff North (because of winds) with a turn to the West. This intersection is North of Runway XX so I would have to cross the Runway XX centerline at some point. Since there was a Aircraft Y West of the airport preparing to land; my plan was to head West and pass behind the Aircraft Y before turning on course. I was cleared for takeoff per my request. While heading West the Tower Controller told me to turn on course and I believe the phraseology was as soon as possible or something similar. At this time our helicopter was not yet behind the Aircraft Y which was approaching fast for Runway XX so this turn on course put our aircraft directly in the path of the Aircraft Y. On takeoff I was given an altitude restriction of 800 ft. AGL which I was well below at approximately 600 ft. AGL. the Aircraft Y was not landing but doing his high speed pass over the runway so its closer rate was extremely fast. What happened was we met in very close proximity on the runway centerline. The distance made me and the medical crew very uncomfortable. In hindsight I should have let Tower know my intentions of continuing West to get behind the Aircraft Y before turning on course. I also could have requested to continue West when the Tower Controller instructed me to turn on course in the flight path of the approaching jet. After the close encounter the Tower Controller apologized to the Aircraft Y pilot.