A supercharger is a belt-driven centrifugal compressor. At low speed the compressor is turning "quite fast" because of the belt running off the engine, and it creates boost. As the engine spins faster the belt turns the compressor faster and more airflow is created, continuing to add power to the engine. This relationship continues until at some point the compressor fan loses efficiency and spinning it "faster" creates "a little more boost," and the supercharger suffers "boost falloff." Think of a little battery-powered hand fan with a 3" blade--it cools your face, and the faster the fan blades spin the more it cools you off--up to a certain point--past that point you can spin the blades as fast as you want and they won't do anything noticeable (beyond cooling your face). You can upgrade to a box fan to cool your whole body, but the bigger fan is going to move more air at lower speeds and cool more than just your face, so there are tradeoffs, including using more energy at low speeds (called "parasitic drain" of the supercharger).
Turbochargers are superchargers, but instead of being driven by a pulley and belt off the engine, the pulley is replaced with a radial-inflow exhaust gas driven turbine (turbo). It's really a turbine driven supercharger, or turbo-supercharger, or just turbocharger. Low exhaust gas speeds don't spin the exhaust-driven fan very fast, so the compressor fan which is on the same shaft also doesn't spin very fast, and doesn't do any compressing. Now you floor it and dump all this exhaust gas into the exhaust fan and start to spin it up, which also spins up the compressor fan and starts to create boost, and that process repeats and repeats and you're waiting on it...this is "turbo lag." Turbochargers don't have power at low RPM (not true any more, but hold that thought for now). However, at higher RPM there's plenty of exhaust gas to spin the exhaust turbine, so there's no lack of compression from the compressor fan at high RPM--meaning turbos don't produce power at low RPM and produce lots of power at high RPM.
From the factory, the Slingshot has "too much" power at low speed, and not enough at high speed (where wind drag slows it down). A turbocharger is the perfect fit, power where you want it. About that lag: Modern turbos are much lighter than the older turbos, so they don't have lots of mass, meaning the lightweight fans spin up quickly. Also, If the turbo is sized correctly, the exhaust gasses at low RPM can be used to spin up the exhaust fan "early" and create boost at low RPM, eliminating lag--the down side is you can floor it at low RPM and create more boost than the engine can handle, blowing it up.
In summary, the turbocharger is perfect for the Slingshot, except you have to be careful not to blow your engine up at low RPM, and since the exhaust and therefore compressor fan doesn't match engine speed a bunch of stuff is needed to get rid of extra boost created in certain situations so your engine doesn't blow up (it's a common theme).
To make things a little more fun, the supercharger guys didn't sit around idle, they worked on compressor fan design, belt ratios, flow characteristics, and got less boost at low RPM and more boost at high RPM, without blowing up engines! Win to the SC guys! Mostly. They still have "boost falloff" up top, but it's not much. For the normal person, the supercharger is the way to go--easy to install, maintain, and won't blow up your engine. For the "I must have all the power I can wring out of this engine at all costs" group, there's still the turbocharger, and all its complexity, which will wring the power out of the top end without remorse. (And blow your engine up if you want--there's that too.)
I have a turbocharger, it's incredible fun, and makes way too much power if you want. It also failed and is under repair. (Things happen.) Squirrel loves the power, but for her we'll eventually put in a supercharger, they give almost as much fun but without all the risk, and they're much easier to maintain with no gauges to keep an eye on, just have fun!