Why did the Persian Gulf shrink during the influence of Ancient Mesopotamia?
There has been lots of works trying to reconstruct the evolution of the Gulf shoreline through time. They do not agree on the exact same dates (see this review by Sissakian et al. 2020), but they generally agree that there was first a transgression (sea level rise) episode following the last ice age, then a regression (sea level drop). Reasons invoked to explain the regression are (not mutually exclusive):

Change in climate. The sea level peaked around the post-glacial climatic optimum. After that, the climate got cooler, which means more ice at the poles and less water in the seas.
Sediment filling of the Mesopotamian delta. The Euphrates and the Tigris carry a lot a sediments, which progressively accumulate in the lowlands, making the Gulf shoreline advance (seaward).
Neotectonics. There are evidence for anticline growth under the area, which could have contributed to its rise above sea level. This activity also changed river paths, which could have promoted point 2.