Why does cold water comes out faster when the hot water is turned off?
In some systems, hot water comes from a first-floor tank pressurised by a cold-water header-tank in the loft. This header-tank also supplies the cold water taps. Part of the reason for this is that mains water supply to the house is at a high pressure, the header tank provides a lower and more constant pressure that is more suitable for household taps and appliances.

Normally the cold water tank has separate outlets for each to prevent the sort of problems you describe. If your system instead uses a single low-pressure feed pipe to both cold taps and to the hot-water cylinder, that might explain the symptoms.

Since you say 'tap' and not 'faucet' I suspect you may live in a locale where the above applies.

A better plumber might be able to advise what can be done to improve the situation.