Why treaty battleships adopted quadruple turrets

Why did so many treaty-era battleships and preliminary designs use quadruple turrets?

They saved weight by sharing one turret's armor and machinery among more guns, at the cost of greater mechanical and maintenance complexity.

Treaty designers needed to maximize firepower inside a fixed displacement limit. Every separate turret requires armor, rotation machinery, handling rooms, and supporting structure, so turret count carries a large weight penalty. Adding another gun to an existing mount does not multiply all of those fixed costs. Quadruple turrets can therefore concentrate a battleship's battery more efficiently than the same number of guns spread across singles or twins. Mechanical complexity and difficult maintenance are the price of that weight efficiency.