First, let me just point out that tanks actually use turbines as powerplants ( the ones that don't use Diesel cycles).
However, for your average commuter car, that would be impractical for a number of reasons.
1. Scaling down turbines changes the Re number, among other things, they don't scale down very well. The efficiency advantage they yield is usually lost when they are scaled down.
2. Turbo machinery required complex parts. For instance, as many on this forum can attest, turbine and compressor geometries can get very complicated. More moving parts, throttling is a problem (in terms of controlling power output), higher powered fuel pumps (to offset combustion chamber pressures), and probably a whole lot of other things related to the increased complexity. This also increases cost (both for purchase and maintenance), which is typically the driving force behind consumer and society's habits.
3. Probably the biggest problem would be this: air intakes would scoop up small animals and shred them to pieces. They would then come out the exhaust system and splatter on the windshields of following cars. This is much more objectionable to simply flattening them on the road, as conventional automobiles currently do.