Meet the skid-steer loader:
It's a versatile beast of a machine, packing lots of power, multiple attachments and (most relevantly to the topic at hand) great maneuverability.
It does exactly what you suggested. It has stationary tires, which turn at different angular velocities in order to turn the vehicle, like a tank. It's easy to turn in a full circle without changing the machine's position. This gives it a number of advantages:
- It can readjust its position when loading and unloading things, if (for example) the forks need to move a few inches to the left or right.
- It can move a bucket sideways to smooth out dirt or gravel.
- Its turn radius is 0, meaning it can get into tight spaces not designed for vehicles.
Compared to a compact track loader (a similar machine that uses tracks instead of tires), the skid steer has these advantages and disadvantages:
- Lower maintenance and repair cost
- Lighter and faster
- Operates poorly if at all in muddy or snowy conditions
- Affects the ground more
A car has little to no need for any of these advantages, as they're not used to load, unload or landscape (though attaching a trailer would be much easier), and generally travel on pathways designed for them.
There actually exist accessories for wheeled vehicles that provide mini tracks for each wheel, to be used in mud or snow:
But even so, the basic steering functionality is still by directed turning rather than skid steering. In fact, there are even machines that are designed for tracks that steer in the same manner:
If even some tracked machines steer in the same manner that cars do, instead of by skidding, there must be some advantage, right?
One advantage (the above curbing machine steers conventionally for this reason) is that it's much easier to form a smooth curve. While computer systems could be used to determine each tire's rotation speed to produce a given turn radius, it's much simpler to simply turn the wheels. Also, as pointed out by @alphazero, one is that skidding reduces traction significantly, which can lead to uncontrollable situations at high speed. There's also the matter of wear and tear resulting from the skidding (and protip, try to turn the steering wheel in your car only when moving, if practical).
Cars generally steer by turning their wheels instead of skidding because few of the advantages of skid steering have much application to a car but the disadvantages range from expensive to complex to disastrous.