The 2.4 has a greater volume of gas exiting the exhaust, but it also needs a greater volume entering the intake to develop the same pressure, so it balances out (mostly).
This is why the K04 runs out of air faster on a 2.4 than the 2.0 however, the amount of air it's outputting is in no way related to the volume (not the pressure of that volume since gas can compress) of air the engine takes in other than how much exhaust gas is coming out the other side. The valves only open so high and for so long regardless of the pressure of the air on the other side of the intake valve. More exhaust volume at a higher speed = greater turbine speed and faster turbine acceleration. You can have that output of the turbo going to something completely different and the volume of exhaust gas exiting the engine is still what dictates turbine speed and its acceleration speed(though far lower since you're not shoving the output of the turbo back into the engine).
If you think about how a turbo works you will see that lag is inevitable, so matter how well everything in working.
Lag is inevitable with a turbo vs a supercharger however...
At steady-state, when no boost is needed, the waste gate is open to allow exhaust gas to bypass the turbo. This reduces the turbine speed, which reduces the amount of air being compressed, and is what makes the turbocharger more efficient than the supercharger. The supercharger always runs in sync with the engine speed, and regulates boost by bypassing compressed air back into the intake, so it is always using energy compared to the "idling" turbocharger that is using essentially none.
The disadvantage of the turbocharger shows up in transition to boost. You open the throttle and the wastegate controller senses the drop in manifold pressure so it closes the gate. This forces more exhaust gas into the turbine, increasing its speed, which pumps more air and builds boost. The time it takes for the turbine to come up to speed is the lag. It can be reduced through proper sizing and better design, but it will always be there. The supercharger only has to close the bypass because the compressor is already at speed, so boost builds as soon as the air in the intake speeds up.
I'm no expert at turbos but I know that to be wrong.
The wastegate is a gate in the turbine that allows exhaust gas to bypass the turbine thus limiting the turbine's speed. That much is true. HOWEVER, the wastegate actuator holds the wastegate closed when the amount of boost being produced is below atmospheric (about 14.7 psi at sea level). Remember, "boost" is the amount of air above atmospheric so if we use the 14.7 at sea level as our atmospheric, 1 psi of boost is 15.7 psi of air pressure. when we're not "in boost" the turbo still compresses the air it sees but that level of air pressure is still below atmospheric.
This is why wastegate actuators are measured in psi. A 5 psi spring like we have in the LNF's K04 turbo will hold the wastegate closed so long as the difference between atmospheric pressure and turbo output pressure is below atmospheric pressure + 5psi so roughly 19.7 psi of total pressure. Once the output pressure of the turbo reaches 5 psi above atmospheric, the wastegate begins to open to control the turbine's speed and keep it running at a speed which results in the impeller producing 5 psi of pressure above atmospheric.
I know this to be true because I ran my K04 on nothing more than spring pressure for a few months before I got my mechanical boost controller.
In the LNF, the wastegate solenoid goes between the compressor housing and the wastegate actuator. If it fails, it fails open allowing the actuator to function just like stated above. However, when operational, the computer reads manifold pressure and tells the solenoid to stay closed until manifold pressure reaches desired boost level. It will then partially open the solenoid to allow some air pressure to get to the actuator so that the targeted boost pressure stored in the ECM can be maintained.
The wastegate does nothing more than control peak boost output of the turbo.
The reason you have turbo lag isn't because the wastegate is closing and the turbine is STARTING to turn but rather the dependent nature of the compressor's impeller speed and the turbine's impeller (I thought the turbine wheel was called something else but can't remember it) speed.
For a given exhaust force (mass x velocity for those who didn't take physics so the amount of force produced by the exhaust is both dependent on the volume or mass of air exiting the engine and it's exit velocity) the turbine will spin at a given speed and produce a given amount of boost. So lets say at cruise the force of the exhaust coming out of the exhaust is 2. Not a real number or any kind of unit of measure but we'll use it to keep my brain from imploding. For this, the compressor produces 1 psi of air pressure...this is 13.7 less than atmospheric so the engine is still running in a vacuum sucking air in and we see no "boost" (because boost is pressure above atmospheric and we're still below). Now lets say at WOT we need the exhaust force to be 100 to reach our targeted boost pressure of 20 psi above atmospheric so we nail the throttle.
Well for one, we haven't burned and exhausted the air that was coming into the engine the moment we opened the throttle. It has to go into the engine, mix with fuel, burn, then be pushed out and travel through the manifold before it hits our turbo. Sure that happens REALLY quick, but its still time. Now that air that is coming out of the exhaust was only at 14.7 psi of total pressure...or 0 boost. So it's not a whole tone of air hitting the turbine.
But, it is more than when we only had 2 so the turbine speeds up when it sees this pressure. But, if you have ever blown into a pinwheel toy, you know that it isn't at max speed as soon as you blow into it even if it is already spinning (and, of course, you are blowing harder than you did to get it to the speed it was at when we start this scenario). It takes a couple moments to accelerate...just like your car isn't doing 100 as soon as you step on the gas...and just like the turbine needs time to accelerate.
As it accelerates, it's speeding up the compressor wheel and that's raising the amount of air going to the engine. This means more air in and thus more air out. Since there is more air out there is a greater mass of air hitting the turbine wheel and thus speeding it up more which continues the cycle. All this happens very rapidly of course but it takes time and that time is why a turbo lags. The wastegate won't come into play until you are outputting a level of air pressure above atmospheric that you have set the boost controller (ECM in the LNF) to regulate to with the wastegate.
The supercharger, on the other hand. Also has a bypass valve but it works very differently. It allows air to bypass the blower/compressor (a roots blower just blows more air into the intake manifold compressing it in the manifold. A twin screw compressor compresses the air in the supercharger and THEN shoves it into the manifold.) and enter the intake manifold directly. This prevents the blower/compressor from seeing air and thus trying to compress it when it doesn't need to (and thus restricting air from reaching the engine from the throttle body and making a crap ton of heat...like enough to burn you when you touch the blower/compressor). When the air pressure in the manifold reaches the pressure of the atmospheric air around it, it causes the bypass valve to close causing the air to pass through the blower/compressor and into the manifold.
Since the blower/compressor is already turning at the speed of the engine and it's air output, or rather it's method of locomotion and the rate of said locomotion, is not dependent on it's air input, as soon as the bypass valve closes air is entering a device that is already up to speed to create boost.
As for how a supercharger (roots or twin screw) control boost, it is by the rate at which they turn in relation to engine speed. You can speed up a supercharger to get more boost or slow it down to get less by changing the pulley size. A smaller pulley speeds up the impellers inside and creates more boost while a larger pulley does just the opposite.
Now as engine speed increases, the speed at which the impellers turn increases. This doesn't cause more boost though since the engine is now demanding more air per minute (since RPMs have gone up) the supercharger is just keeping up with that demand by producing the same boost level more frequently (because it's RPM has gone up relative to the engine)
So the following is correct:
It doesn't matter what the engine speed is, the rotor in an idling turbocharger is not going to be turning as fast as one in a boosting turbocharger. It is going to have to speed up to go from one state to the other, and speeding up takes time.
Now, the one thing that CAN open on a turbo charger when there is no boost is the Bypass or Blow off valve. However, the system between the turbo and throttle body must have a pressure higher than what is in the manifold (which is after the throttle body) in order to open. But this only happens when the throttle closes causing a vacuum behind it in the intake manifold while the charge tubes and intercooler still have air under boost pressure in them. This is to prevent the boosted air from flowing back into the turbo and slowing down the compressor wheel which can damage the turbo. The Bypass or Blow off valve releases that pressure and this has nothing to do with how quickly the turbo builds boost.
Again, I don't claim to be an expert in each but this is what my I have absorbed with my Supercharged Grand Prix (M90 and Whipple) and turbo charging the Sky.