Haha as a car enthusiast, whenever I see "turbo" my next thought is always the inherent downside, "turbo lag" (the non-zero time it takes for the turbo to actually kick in)!
Split the turbocharger into a recovery unit, genset, a supercharger and a battery and no more lag.
There are better things you could do to increase engine performance that are lighter, smaller, and cheaper.
Also of you really really want to defeat turbo lag. The easiest way is to seriously delay ignition timing so the fuel is still burning as it enters the turbine leaving more energy for it to extract and therefore staying spooled up.
edit: Not to say your idea wouldn't work, indeed I'd love to see it. Its just not a practical solution. Not that In ever let practicality stop me.
Edit to the edit: https://dieselnet.com/tech/engine_whr_turbocompound.php
https://www.eere.energy.gov/vehiclesandfuels/pdfs/deer_2004/...
So, ya know, disregard everything above.
Part of it is that there’s another reason besides a turbo to have a little extra energy left in the exhaust: thermal catalysis of noxious chemicals in the exhaust flow. So there’s a bit of unburned fuel going to the cat for emissions control and people have wondered if we can take some of it back after the converter. For a while every time there was a breakthrough in solid state heat recovery someone mentioned vehicle exhaust.
Like, producing 1 bar of boost for a 2L engine is somewhere between 30-70 hp! Turbos extract that energy from otherwise wasted, extremely hot exhaust gases, which is what makes them efficient. That's also what made them an improvement over supercharging for the majority of applications, because in a supercharger that energy cost comes direct from your usable power.
What "fixed" turbolag in any engine that isn't an 8L diesel is things like variable geometry, lightweight turbines, better bearings, better engine management, and just smaller turbos for the same boost.
It's also massively impacted by driving behavior. My GTI with a teensy tiny turbo still has a half second or so of lag if you try to go from no throttle to full throttle, like if you are really bad at "launching", but if you know what you are doing you will be holding some revs such that the ECU can slap open the throttle plate and dump fuel in just a few cycles and get full 1 bar of boost by the time your clutch is fully engaged. I have done this accidentally more times than I would like to admit, which tends to spin the tires :/
Formula 1 adopted the exact system you are describing, but it's stupid expensive for minimal benefit and I would bet is used more for energy recovery from waste exhaust gasses than actual lag reduction, since F1 cars should be able to almost never be in a regime that would result in lag, and if you have spare joules, you can just throw them directly at the wheels instead of the turbo. It's called the MGU-H system.