A twitchy rear isn't normal for Miatas usually. As others have said, probably a bad alignment and/or crappy tires.

And there's no reason to do these difficult swaps...SR20's and rotatries and so forth. The 1.8L Miata engines respond very well to boost, and there are a bunch of cheap turbo and supercharger kits out there.

So yeah... Miata + turbo + coilovers + good tires = Very Yes.