To allow clearance for those tall wheels to turn.

 

That almost sounds right.

 

Probably technical. Back then, engine covers weren't as wide; they didn't have to be, since most engines were straight fours or straight sixes without power accessories and other options. They were narrow at the front because most makes put the free-standing headlights between the radiator and the fenders, but widened toward the cowl. To make the sides even, the front of the passenger compartment had to be as narrow as the cowl, but the body could widen somewhat toward the rear. Cars had running boards well into the 1930s, so neither the front nor the rear seats were as wide as they'd become after fenders evolve into quarter panels.

 

The Model A Fords and many others had a mostly flat frame. They would 'drop' the bodies onto the chassis and bolt it together. Then add fenders, running boards, hood, etc. The floorboards were ~ 20 inches above the ground, so running boards were a necessity. But they limited the width of the body, especially the front. The Model A production was pushed by Edsel Ford, not by Henry Ford. The body manufacturers were known as coach builders.