But how is the water evaporating only slightly slower than I can pump it into a roofed off reservoir? Is the water wheel's position the problem? Or is the water evaporating out of the part of the river I channeled?
Edited for complete re-working of response:
There are two issues.
1). Evaporation.
2). The fact that loose water doesn't care which way you want it to flow.
Water pumps in from below, sloshes down a corridor (lowering in depth the whole way), and then reaches your reservoir. From this point, each individual 1/7 of water will flow at random in any direction it feels like. Therefore, it spreads out. Now, every tile of ground in the "dwarves-stand-here" section at the top of the reservoir is space for water to evaporate. Because it no longer has any compulsion to head straight North, it can start to vanish in the reservoir before it even reaches your target.
When you open the floodgate to the farm, you're hoping that some units of water will happen to head that way, but you haven't allowed the reservoir to fill up enough that any noticeable amount of water gets through. Flooding areas on the same level as your water source (i.e., the pump) is tricky this way. The solution is to allow your reservoir to fill more first, so no tile gets down to 1/7 (the threshold for evaporation), and ideally is at 7/7.
The waterwheel only providing intermittent power causes another problem, as it is slowing down the flow and giving you even more time to lose water.
Solution:
Close all floodgates. Flood the area next to the pump completely. Flood the reservoir next. Then open the floodgate to the target.
Long-term ideas:
Drain completely. Remove floors in reservoir. Start again. Also move the waterwheel. Also consider a water tower above your target, so as to pressurize water. Also learn how to deal with pressurized water before you try this.
Caveat:
I don't actually know how critical the floors are, but I'm fairly certain that water only evaporates when standing on a ground tile, not other water. I do know that the stated solution will work.
--Rexfelum