All travel in an arc, but for light fast bullets this is negligible at most distances. For slow heavy bullets the arc is more pronounced so it does become a targeting issue. (Weight actually doesn’t matter, it’s just speed ... but a fast heavy bullet will produce intense recoil.)
An 81mm shell has a pronounced arc of travel that lets you pound positions on the reverse side of a hill that are unassailable by 90mm tank guns. Just sayin’ . . .
In fact all bullets drop at the same speed. The same speed they drop when you hold one out to your side and release it.
The only question is how far the bullet traveled in that period of time.
Gravity is a constant and is pulling the bullet down as it is in flight at that constant rate. If you drop a bullet from shoulder height (5 feet), it will take 0.56 of a second to hit the ground. If you fire a .45-70 at 1350 ft/sec assuming a perfectly level rifle, it will go forward about 250 yards in that time. All the while, gravity is pulling the bullet down so that it will arrive at the ground in 0.56 seconds. In other words, the bullet will drop 60 inches over 250 yards, which lines up almost exactly with the trajectory chart of a .45-70 shooting a 530 grain bullet at 1350 ft/sec. There is some rounding error in my math.
If a rifle propels the bullet at a very fast 4,000 ft/sec, it will travel almost 750 yards in that 0.56 seconds resulting in a much flatter trajectory.