If an object enters the water, some air is dragged behind it ad the object breaks the surface of the water. There’s also a thing called cavitation that affects underwater propellers (and probably bullets as well): a fast moving object can “rip” the water apart, creating a bubble (cavity, hence the name) filled not with air but with vapor from surrounding water. These bubbles collapse quickly but they are still visible.
However, don’t look into Hollywood scenes too much, most of the bubbles are just added for visuals. Actual propellers are made to reduce cavitation since it can damage the blades. Maybe some submarines have some sort of exhaust near the propeller but I won’t make any guesses here.
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