A narrow pipe makes water come out faster
Continuity, and why squeezing a flow has to speed it up.
Water forced through a narrower opening comes out faster because the same amount of water has to pass through every section of the pipe in a given moment, and the only way for a smaller opening to keep up with the flow arriving from a wider one behind it is for the water inside it to move through faster.
Why the same volume has to pass every point
Picture the water moving through a pipe as a continuous, unbroken supply, with no gaps forming anywhere and no water piling up and going nowhere at any point along the route, which is a fair description of an incompressible liquid like water filling a pipe completely. Whatever volume of water passes any one cross-section of the pipe in a second has to be exactly the volume passing every other cross-section in that same second, since water cannot vanish at a narrow point or pile up there indefinitely. If the pipe narrows partway along its length, the cross-sectional area available for the water to pass through shrinks, and the only way the same volume can still get through that smaller opening in the same amount of time is for the water to be moving faster there than it was in the wider section behind it. This relationship, the width of the opening and the speed of the flow trading off against each other so that their product stays constant, is the same continuity idea that sits underneath Bernoulli's principle and is often stated as a simple sentence in its own right, the flow's speed multiplied by the pipe's cross-sectional area stays the same all along its length. Narrow the pipe to half its area and the water in that narrowed section has to move at twice the speed to keep the same total volume flowing past every second. Nothing pushed it harder from behind; there was simply no other way for the same supply to keep making it through a smaller gap. Widen the pipe instead of narrowing it and the same reasoning runs the other way, water arriving into a suddenly larger cross-section slows down, spreading the same volume more thinly across the extra room it has just been given, which is exactly why water gushing out of a narrow tap slows visibly the moment it lands in a wide basin and is free to spread across a much larger area than the tap opening ever offered it.
Half the width, four times the speed
Halving the width of a round pipe quarters its cross-sectional area, since area shrinks with the square of the width, so the water passing through the narrowed section has to move four times as fast. Even a narrowing that looks modest to the eye asks for a large jump: a pipe squeezed to two thirds of its width leaves less than half the area, and the water there has to go a little more than twice as fast as it did upstream.
A thumb over the end of a hose
Pressing a thumb part of the way over the open end of a garden hose, narrowing the opening the water is allowed to escape through, sends the water shooting out in a fast, forceful jet that reaches across the lawn, where the same hose left fully open only splashes onto the ground a stride or two away. Nothing about the tap or the pump supplying the hose has changed, the same volume of water is still arriving through the hose every second as before, only now that same volume is being squeezed through a much smaller opening at the very end, and the water has no choice but to speed up to get all of it through in time. Removing the thumb restores the wide opening and the jet immediately slows and spreads again, even though the supply behind it, the tap and the hose itself, never stopped delivering exactly the same amount of water throughout. A narrow kitchen tap nozzle, a garden sprinkler head, and even the narrowed neck of a bottle being poured all rely on the identical relationship, restricting the opening a fixed supply of liquid is allowed to pass through and letting that restriction do the work of speeding the flow up rather than needing any extra force applied further back.
Nozzles, gorges and oversized drains
The rule that a narrowing speeds a flow up and a widening slows it down, with no pressure trick needed to explain either, accounts for a wide range of everyday plumbing and irrigation choices. A hosepipe nozzle sold specifically to produce a powerful jet is really just a carefully shaped narrowing, sized to squeeze a garden tap's ordinary supply through a small enough opening to produce useful reach without needing a stronger pump behind it. A river that appears to speed up dramatically where it passes through a narrow gorge, after moving sluggishly across a wide open plain just upstream, is obeying the identical relationship on a far larger scale, the same total volume of water squeezed into a much narrower cross-section having no option but to move through it faster. And a drainage pipe deliberately sized wider than strictly necessary is often built that way specifically to keep flow speeds low and manageable, since a narrower pipe carrying the same volume of water would force that water through at a speed the pipe or whatever it drains into might not be built to handle safely.