The following is not a very technical explanation - and involves a gross oversimplification of the way that fluid flows through pipes - but it may help:
The inner surface of the pipe presents resistance to the water flowing through it - tending to slow it down. That resistance is along its entire length, so a 10m long pipe has twice the resistance of a 5m long pipe.
A pipe's circumference - where the pipe is touching the water - is proportional to the diameter, but it's cross-sectional area - the volume for the water to flow through - is proportional to the square of it's diameter.
Finally, the resistance of the pipe to the flow of water increases with the rate of flow.
For simplicity, if we assume that one pipe is double the diameter of the other, then:
Say diameter of pipe1 (the narrow one) = d
Area of pipe2 = (2*d)
Circumference pipe1 = PI*d
Circumference pipe2 = PI * (2*d)
Area pipe1 = PI * (d/2) * (d/2) = PI * (d*d) / 4
Area pipe2 = PI * (d*d)
So...
If we double the pipe diameter for any given length, the area of the inside surface doubles but the volume quadruples - so overall the ratio of volume/surface area doubles. Thus, in a larger pipe, less of the total water in the pipe is in contact with the frictional surface so the water can flow more freely.
Not only that, but for a given flow rate of water, a big pipe means the water can travel more slowly, so the resistance to flow caused by the surface of the pipe is reduced.
To relate it to your question, the increased resistance of the narrower pipe (compared to a wide pipe) will act on the water all along its length, and will reduce flow rates.
I hope this helps!