"Turbines"?
You mean Gas Turbines?
Well, they have been tried, several times between the '50s and '80s.
I'll never forget the case of the Grand Touring class racer where, on the grid of a race it was necessary for the marshals to move the car forward - because the exhaust was charring the paint on the front of the car in the row behind, at "idle" please note... Now, imagine parking something like that in a tight space!
Black humour aside ('Charring'. Sorry, it's a crap job, but someone has to do it, right?) , Gas Turbine cars have a horrific thirst for fuel compared to conventional cars - even compared to the real gas guzzlers - when asked to constantly change operating RPM.
Another issue is that Gas Turbines have an outstandingly poor throttle response - worse in fact than the worst, which means, these days, the very first, turbocharged conventional engines. Hardly the sort of thing you need when pulling out onto a busy roundabout for example... And I say this as the driver of a modern auto-gearbox turbo car who had to learn the art of anticipation under those circumstances.
Which brings us to another interesting point. Thus far, all Gas Turbine powered cars I can recall have not actually had a gearbox, auto or manual (other than a simple 'reverse' gear which simply reversed output rotation, rather than changed transfer ratio and a simple reduction gear).
This because of the torque response of a Gas Turbine, which resembles that of an electric motor - think milk float - they all used nothing but a torque converter to handle the transition from stationary to moving, then depended on the very high RPM range (from idle (12k rpm anyone?) to 'OMG! Do these things REALLY rev that high?') to handle the rest of the speed range.
And the power/toque sapping torque converter is the achilles heel of the classic automatic gearbox, which is why the trend lately is toward 'automatic' gearboxes which are in fact what amount to manual gearboxes with automated gear selection and electronically controlled clutches...
These are far better suited to the power/torque characteristics of a 'conventional' engine (whatever THAT is these days) and pretty much useless with a gas turbine.
Drivable cars need the ability to produce varied torque/power output in rapid response to changing requirements. Gas Turbines are far more suited to running at a steady speed. Which is why they are used in large scale electricity generation. And why aircraft manufacturers tend to emphasise 'cruise' performance. Gas turbines really aren't good (noise/pollution/fuel efficiency) during takeoff, climb and, especially, 'maxed out' (excluding after-burn/military thrust conditions, under which they burn fuel like one of those well-heads Madman Hussein set fire to...).
However, there does exist one possibility - there are such things as small Gas Turbines. Very small. Right down to model aircraft size.
A possibility which springs to mind is a vehicle with a small Gas Turbine that only drives an electric power generator. Running at a fixed speed (cruise mode - highest efficiency - minimum emissions). The vehicle is then propelled by electric motors - which DON'T require torque converters. Add the usual near obligatory stack of Lithium batteries, regenerative braking and a good control system and you might just have the basis of a rather nifty car...
OK, it's only an idea. A fairly complex one, but I doubt there's any simple solution.
So, someone shoot the idea down. I'd seriously like to see what holes the concept has.