Gravity acts on both bullets the same way. It acts on the fired bullet the second it leaves the gun barrel. In the absence of air, both fall at the identical rate. Cite.
Common sense may tell you otherwise – and common sense is often wrong – but gravity doesn’t wait to act just because a body has a rapid forward motion. It’s there all the time.
If you were to ignore the forward component of the fired bullet, and charted its vertical motion vs. time, this chart would be identical to the dropped one, in a vacuum, of course. Both fall the same distance in the same time.