I own a ICE car with a manual gearbox. The lag is there. The engine needs time to find his torque at low RPM so you need to drop one or two gears if you want some power. Mine doesn't have a turbo, but when I drive manual cars with Turbo, the turbo lag is huge. A manual gearbox doesn't fix this problem. You may drop gears before you need the power, to overtake someone for example, because you plan ahead and want the full power, but that's slow.

And even when I do it fast, I'm much much slower to manipulate the clutch and change gears compared to a modern automatic gearbox.