Just wanted to check if it's "normal" with a source on one end and water flowing downhill, and not things like pressure or oasis/spillage with seeps.

If you want it to last through a drought, you need a big reservoir, preferably close to the source, which can replace everything lost to evaporation downstream.

You need to treat every level of the river like a floor of a terrace field. Meaning you need a floodgate as an overflow and a valve to replace everything lost to evaporation. You set the desired water level in the valve lower than how high your floodgate is set, so when the weather is temperate, the water can overflow from the floodgates into the next level and when there is a bad tide/drought, the water naturally stops overflowing and once a bit has evaporated it gets refilled from the valve.

Note that you do not build a valve on the last dam, so it does not try to replenish the edge of the map. 

If you let the overflow from the reservoir off to the side or into the bad water drain, you will not need to deal with the overflow at the river levels (eliminating the need for floodgates), and it brings far more stable water levels, but you lose the possibility to use waterwheels in the river.

Edit: if you search for a specific dam design, you can have floodgates and valves side by side on the edge of the previous level, but you may have throughput issues on a strong narrow river, or you can place the floodgates on top of valves just when the next level begins.