Did you not just contradict yourself?
You say that "bullets never curve" which is generally true, except for some circumstances (wind, coriolis effect), but that's beyond the scope of this. You even state that bullet fly perfectly straight, which is also true to some extent, depending on your point of reference - bullets fly in a ballistic arc, which is not a straight line, but can appear so from the view of the shooter. However, 'straight line' can be considered as true for x plane movement - the bullet doesn't go magically to the sides, while it goes up and comes down on the y axis in an arc.

Then a few sentences later, you claim that the projectile needs to care about the velocity of the shooter, which can only be true if the projectile does curve.
So which one is it? It can't be both.
Your velocity (or to be more precise - the velocity of the weapon firing the bullet) does not affect the bullets trajectory - the bullet will always fly in a ballistic arc towards the aiming point, the lateral or vertical movement of the gun in question has no impact on it if you keep the aiming point at the same point - adjust your aim. If you don't keep the aiming point fixed to the same point, then it obviously will change with the movement of the gun.