It's not that hard to explain. When the rear tires (friction forces) push on the ground, they also push on the front tires (rigid body constraint through the chassis, axles, etc.). The front tires are turned at an angle, however; so unless you're doing something unsafe, the friction forces on the front tires constrain them and the motion of the entire car. This constraint is only perpendicular to the front tires, however, and they, as well as the rest of the car, are free to move along the angle they are turned to. So the car turns. If you're doing something risky given the driving conditions, however, the car might skid, tilt, or even flip over.
 Quote by Bashyboy
Well, how does a tire rotate, wouldn't it have to overcome some resistive force?
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The tire rotate because of the axle, the crank, and ultimately the combustion of gasoline inside the cylinder.
 Quote by Bashyboy
Okay, that sounds good. So does the friction have an equal and opposite force? Or do not all forces have to have equal and opposite forces?
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In Newtonian mechanics/model, all forces are paired. The friction pushes on both the tire and the Earth, but in opposite directions.