Why a smaller gun need not produce a much smaller turret

How much smaller could an Iowa-style turret become if it used a hypothetical 15-inch/50 gun?

Only a little; the recoil and ammunition-handling systems would still need substantial space, so the same housing could simply be roomier.

Replacing a 16-inch/50 gun with the proposed 15-inch/50 weapon would not greatly reduce the required turret space. Shell handling, hoists, and recoil machinery occupy much of the enclosure. The difference in shell diameter would leave those systems broadly similar in scale. Retaining the existing housing and accepting more working room is therefore suggested as the practical result.