Fact: Magnitude of upthrust is equal to the weight of fluid displaced.
When the cargo was in the boat:
Say the boat was on the right. Some people may think that the right side is heavier because of the additional weight of the boat and cargo. Other people may think that the right side is lighter because of the weight of missing (displaced) water.
Guess what? The missing water weighs exactly as much as the boat and cargo!!! Because the boat floats, the weight of the displaced water (which equals upthrust in magnitude) must be equal to the weight of boat and cargo. (Archimedes Principle).
So the missing water exactly compensates the weight of the boat and cargo. The weight is evenly distributed along the bridge.
When the cargo was by itself:
When the cargo was sunk on the right, the right side again has the additional weight of the cargo but less water.
But this time round, the missing water does not weigh as much as the cargo. The fact that the cargo sank meant that the weight of the displaced water (which equals upthrust in magnitude) is less than the weight of the cargo.
So there isn’t enough missing water to compensate the additional weight enough. So the right side is heavier, and the bridge toppled.
P.S. By the way, water bridges really exist. The Romans built them to transport water and boats. See aqueducts.