I have to ask, do you know how turbos produce power? The gearbox has nothing to do with it.
Turbos are spooled up by the exhaust gases and gains speed as the exhaust flows faster. As the turbine spins it compresses air and shoves that air into the engine. That cant happen until the turbine is spinning and it cant spin until the gases are flowing. Lag.
There are ways to eliminate lag;
Anti lag - in a nutshell it spits fuel into the turbo where the heat explodes it, which causes the turbine to spin, etc etc. This dramatically decreases the life of your turbo. Incredibly expensive overall.
Fit a sequential turbo, or twin turbo - a smaller unit in front of the larger one. It spools up sooner and when it runs out of puff the larger one takes over. Again expensive, plus needs more physical room in the engine bay.
Twin scroll turbo - they have two inlets rather than one and do fancy stuff that gets the blades spinning sooner.
Fit a supercharger in front - the turbo works from exhaust scavenging whilst the supercharger works from the engine power itself; it is powered by the crank. The effect is instant with the downside of less power overall as the supercharger is sucking some of it to run itself. Have both for fun times.
NOx - you will blow up your engine. But it does work.
There may be other ways.
Or you could learn to drive with lag in mind.