Water losing pressure along a pipe is losing it to friction against the wall, and the relationship between pipe diameter and that loss is far steeper than it looks.
In the Hazen-Williams formula, the standard method for water distribution, diameter appears to the power 4.87. That exponent is the whole story.
Halving the pipe multiplies the loss by twenty-nine
Push 5 litres per second through 50 metres of 100 millimetre pipe and you lose 0.263 metres of head. Push the same flow through 50 millimetre pipe and you lose 7.702 metres.
That is 29 times the loss for the same water, over the same distance, in the same material. Two to the power 4.87 is about 29.2, and the calculation is doing exactly what the exponent says it will.
Nothing else in the system behaves like this. Doubling the length doubles the loss. Doubling the flow multiplies it by about 3.6. Only the diameter has a near fifth-power grip on the answer.
One size up is usually the cheapest fix
Take 10 litres per second over 100 metres. In 100 millimetre pipe that costs 1.902 metres of head. Step up to 125 millimetre, one size, and it falls to 0.641 metres, a two thirds reduction.
A pump big enough to push through the undersized pipe has to be bought once and then paid for every hour it runs, because it is doing avoidable work against friction. The larger pipe costs more once and then costs nothing.
The Hazen-Williams calculator makes the comparison directly: enter the flow you need and try the diameters available.
Roughness matters, and it gets worse
The C coefficient describes how smooth the pipe is. Plastic is about 150, new steel about 130, badly corroded cast iron about 80.
At 5 litres per second through 50 metres of 100 millimetre pipe, C of 150 loses 0.202 metres and C of 80 loses 0.647. That is more than three times the friction for carrying identical water through identical geometry, and it is the reason designers specify a lower C than the pipe currently has: the pipe ages, and the system has to still work in twenty years.
Velocity is a second constraint
Head loss is not the only limit. Most design standards keep velocity between about 0.6 and 3 metres per second. Too slow and sediment settles out; too fast and you get noise, water hammer and erosion of the pipe wall.
Oversizing to eliminate friction can push velocity below the sedimentation threshold, so the answer is a range rather than simply the largest pipe affordable. The calculator reports velocity alongside the head loss for this reason.
Fittings are not in this number
Everything above is straight pipe. Bends, valves and tees each add their own loss, usually handled by adding an equivalent length of straight pipe or by a K-factor method.
On a long straight run these are a minor correction. On a short run with a dozen fittings they can exceed the pipe friction entirely, which is worth remembering before concluding that the pipe is the problem.
For the physically general method that works on any fluid, see the friction loss calculator. For the friction factor it depends on, see the Darcy friction factor calculator.