I have often heard of cases where dams reduce the flow rate of rivers downstream. This makes sense in cases where some of the water that is captured by the dam is used for irrigation and such, but in some cases, dams are built solely for hydro power and flood control, and my impression is that those dams still reduce the volume of the river downsteam. Am I wrong in this assumption? Otherwise, what is the mechanism by which dams that are solely used to delay rivers momentarily cause their rivers downsteam to lose water? I could see a small increase in water loss due to evaporation that can be attributed to the increased surface area of the reservoir that the dam creates, but I would not imagine this factor to be that significant.
The answer has nothing to do with Bernoulli's equation.
A large dam across a steep canyon will define a large catchment volume. It is not uncommon for such a catchment to take as long as ten years to fill to capacity. Even if some of the stored water is spilled each season to furnish some downstream flow, as long as the catchment is in the process of filling, it is removing water volume from the river flow and the downstream flow will necessarily be reduced.
Politically, it is also common to overallocate the stored water among its users- once the water starts becoming available, everybody wants a piece of the action. This further increases the fill time for the reservoir and applies great pressure on the dam operators to reduce downstream releases. Note that since water demand only increases over time while precipitation does not, the problem of overallocation only gets worse with time.
Finally, note that a long river with a lot of elevation drop will support multiple dams along its course. This compounds the time-to-fill and overallocation effects and can actually reduce the net flow downstream of the last dam in the series to zero during dry years. Or decades.
Some dams are designed to even out the natural river flow so that the level or height of the river is relatively constant.
The natural change in rainfall over time causes a natural change in river height which is fine for nature but, since we have built in many of the floodplains that nature designed then people get upset when they get flooded.
I suggest that this is not an engineering problem, but a political one. Yes, dams reduce flow downstream when they are filling up, but this isn't a long-term issue. The issue is that once a dam is put in, the dam owners control the flow, and those owners may have different priorities than people downstream. Those priorities include flood prevention, recreation, power generation, and potable water supply.
So it isn't the dams that deplete water downriver. It's the owners of the dams.
Evaporation is a substantial factor in arid locations like the US southwest. A very large surface area reservoir in a very low humidity causes a lot of water loss. I remember reading that Glen Canyon reservoir was taking several more years to fill than calculated because of evaporation losses.
Studying the water head changes in the system would tell you the reason, I believe while some of the kinetic energy is being transformed into electricity, sound and heat etc.
However, the tailwater (downstream) is controlled by the spillway. And this is how they design the water flow rate which influence the downstream.