Pressure versus flow
Pressure is the force behind the water; flow is how much arrives per minute. A mains supply can be high pressure but low flow if the pipework feeding it is narrow or partly blocked. Fill a bucket for one minute from the kitchen cold tap: under about nine litres suggests a flow restriction worth investigating.
The cheap causes worth checking first
A partly closed internal stopcock, a partly closed service valve under a basin, a blocked tap aerator, or a clogged shower head account for a large share of complaints. All four are free or nearly free to correct and should be ruled out before anyone quotes for pipework.
Whole-house causes
Older properties may still have a narrow supply pipe from the main, or galvanised sections that have furred internally. Gravity-fed systems suffer where the cold tank sits too low relative to an upstairs shower - the head is simply insufficient. Neither is fixed by changing a tap.
The fixes, from cheapest up
Clean or replace aerators and heads; open every valve fully; replace failing flexible tail hoses, which restrict badly when they collapse internally. Then consider a shower pump on a gravity system, or an upgraded incoming supply where flow at the stopcock itself is poor. Upgrading the supply is the only fix that raises the ceiling for the whole house.