Perhaps the bullet's trajectory is tied to a specific point, and when the hero moves, the point changes along with the hero, resulting in a change in the bullet's flight. Maybe it's worth breaking the connection with the parent after the shot?
The bullets are physical (Rigidbody2d) and follows a straight line in the game world independent of the helicopter. The problem is partially the trail render and that in a side-scroller like this you stand still and everything else moves towards you.
