The crux of the matter seems that we modern cars the key battle we are seeing is an efficiency battle and not a power, weight or thrust challenge. Ok, the losses reduce with weight somewhat but modern piston engines are pretty light anyways and there's other weight cutting targets in a car.
So far as I understand, the selling point of turbines is their massive power to weight ratio. Not their inherent fuel efficiency.
Ergo, I don't see what the motivation would be to fit a turbine to a car.
As a practical matter, if turbine technology does advance in any specific way that makes it more suitable for vehicular sources I'd expect the trickle down to be from larger vehices first. i.e. You'd see more widespread adoption in something like train engines where the thrust levels are much larger and gearboxing is not an issue since they typically drive electric motors anyways for traction.
I knew GE had a bunch of turbine running locos in the 60's but they never really caught on. And till a turbine makes sense for a train I see no reason it will for a car.
In niche uses like the Abrams Tank (i.e. huge power requirement, efficiency a secondary concern, space comes at a premium, ease of maintenance needed, robustness) yes we do see turbines.
Even in shipboard applications although there's some turbine use but if they really got efficient you'd see the Emma Maersk run its props on turbines instead of that huge monstor of a piston engine.