They’re wide because the bottoms need to be wide.
The bottom is a pool of hot water evaporating/cooling off. It’s not a chimney forcing out smoke.
The water needs to cool so it can go back into the water loop and get heated back up by the nuclear reactor to drive the turbine which generates electricity.
The towers could be perfectly cylindrical but are instead hyperboloid. I’m not going to guess if that does something special to the air currents but I do know that it is easy and strong to build a hyperboloid with just straight beams: https://en.m.wikipedia.org/wiki/File:3D_dual_use.jpg
Those beams are straight. They form a curved surface though. When scaled up you can build curved cooling towers using strong simple straight steel beams as a skeleton.
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