If it's one of those little Tornados, it won't do a damn thing. And if it's a legit turbo kit, with the re-routing of exhaust gases to spin up the turbo and create boost, you'd probably be able to get a significant amount more power than the 2.7L, but if yours is a CVT I dunno if it would handle the extra power.
The 2.4L in the Compass is one of the world engines (incidentally, so is that 2.7L from the Sportage), and so its very, very similar to the engine you'd find under the hood of a Neon SRT4 or Caliber SRT4. Those are both capable of 300+ hp, so if someone was enterprising enough, they could definitely get those kinds of power in the Compass. It's just be a pretty large cost, especially considering you can get low mile Caliber SRT4s for like $13-14k