Why don't dams overflow?

You’re missing something.
Dams have valves that release water when opened. Dams don’t just block the water. The water flow thru the releases generate power.

It has happened. most modern dams have a spillway that allows high water to flow in a controlled way thereby keeping water below top of dam. See most photos of Hoover Dam, Nevada show the spillway spraying into the downstream canyon.

#1 Dams do overflow
#2 There are spillways for the water to be diverted when it gets too high
#3 The dams are much higher than the water they are intended to catch
#4 Dams do overflow

@YARNLADY And sometimes, just in case you were starting to sleep well at night in the shadow of the local dam, they break.

GQ! I’ve always wondered about this. Thanks Trypto for asking it, and flimflam for the answer :)

Hoover dam, releasing backup water

@Nullo Yes, they do – I am lucky enough to be on higher ground, but there is a major stream a few blocks from here, downstream of a dam.

If you just build a giant concrete wall in front of the water, the water will overflow. Dams allow water to go through them while generating power. They also have spillways that release any extra water at a controlled rate. The hight of the lake formed by the dam can be controlled by the spillway.
As proof, look at the water coming from a large dam. The river doesn’t disappear when there is a dam in it. This river is where the water goes from the lake, instead of overflowing the dam.