For an answer, let's go to Africa's Lake Victoria, which is a major source of the Nile.
It's surface elevation is 3,717 feet above sea level, and the Nile "debouches" (I always wanted to use this word) into the Mediterranean Sea at, of course, 0 feet above sea level. If you drop an object 3,717 feet through a vacuum, it will reach a downward speed of about 500 feet/second, or about 300 miles/hour. Air wouldn't slow it down too much, to about a couple hundred miles/hour. Clearly, the potential energy of 3,717 feet does not cause the Nile to hit the Mediterranean Sea anywhere close to these speeds, either vertically or horizontally.
You'll probably find the reason why if you rub your hands together .... friction. All along the way, as a river drops in height and speeds up, it increases its friction along its banks and bottom and between its various parts moving at different speeds and directions (and, I suppose, even a little bit between the river and the air above it). All this creates friction, and the friction creates noise and heat. The sonic energy is obvious at waterfalls and cataracts and other places where the water makes noises, but the thermal energy is not so obvious. The heat would be absorbed by the components involved in the friction -- the water, river bank and air -- and the water would transfer its heat to things in it, such as fish, and also outwardly in all directions, including, I suppose, outer space.
The friction would slow the river, which keeps the Nile from hitting the Mediterranean Sea at a couple hundred miles per hour. What you'll find is that the river's average speed through one of its sections is related to the river's drop through that section. Rivers tend to be steeper, faster and straighter in the mountains and shallower, slower and more meandering closer to the ocean or similar bodies of water. Ever see a river end in a waterfall into the ocean? Probably not. Over time, the speed of the water over the falls would erode it and the river bed behind it until it reached a much shallower angle.
Another use of the friction energy is the erosion of river banks (usually in the mountains caused by the speed of the waters there), and the deposition of those soils as the river slows down in its shallow approach to the ocean etc. Transporting these solids in river water requires kinetic energy, and when the river slows and this energy lessens, the particles begin to fall out, causing deposits such as the Mississippi River delta, or the extensive river bed of China's Yellow River along the plains leading to the Pacific. With enough deposits, the river will overflow its banks and begin another way along the shallow slope.
The Yellow River
has had a very long history of erosion and deposition, which is why people gave it that name. At its worst, scientists have measured 44 pounds of soil in one cubic foot of "water" that would be 62.42 pounds per cubic foot if it was pure water. The Chinese say that once you fall into the Yellow River, you never get clean again. On the plains, deposition has raised the river bed, and the Chinese have had to raise the river banks to prevent it from overflowing. In some places, the river bed is higher than the surrounding countryside.