A turbo is a fan inside a small volume chamber. It is powered by the exhaust gases of the engine. If it is turning slowly, it is filling and emptying that chamber slowly. when it is turning a few hundred or a few thousand rpm as the car idles it is pumping very little air into the engine. When the turbo is turning many thousands of rpm, it is moving a lot of air into the engine. The more air the engine has, the more fuel it can burn, and the higher the pressure of the exhaust gas turning the turbo fan. When the engine accelerates, it takes the engine some time to rev up, and to start generating a lot of exhaust gas to turn the turbo fan. As the turbo starts turning faster, the engine exhaust helps it to turn the turbo even faster, which helps to turn the engine faster, etc.
But it takes a bit of time for the turbo to take advantage of the initial increase in engine exhaust.
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