First you have the wheels on the car. We can consider wheels as devices that give us arbitrarily low forces in one direction (perpendicular to the axis when the wheel is rolling) and arbitrarily high friction parallel to the axis.
Next you have a vehicle with two axles, where the front axle and rear axle are at an angle.
As the car is rolling, this creates forces at different points on the car, which becomes a torque. If we imagine the car is rolling forward with the front wheels to the left, frictional forces arise that force the axle to the left.
This unbalanced force on the front of the car has two effects:
- Accelerates the car to the left
- Turns the car in the leftward direction.
Assuming the wheel angle is held constant by the steering mechanism, this continues as the car moves forward. The car is constantly accelerated to the left (in the car's reference frame), but the turning of the car changes the angle so the car moves in a circle rather than accelerating off in one direction.
(Once the car begins turning, the torque is zeroed out by the rear wheels. If the rear wheels did not do this, the car would spin rather than turn)
Why does it move in a circle? Why does the wheel change rolling direction? Why does the force from the road change direction?
We are assuming there is some mechanism that holds the "turning" wheels (the front wheels on a normal car) at some angle to the body. In the normal case, it is the driver holding the steering wheel at some angle. We could replace this with a block of wood where the front wheels are locked at some angle.
With this true, the torque on the vehicle is transmitted to the driving wheel, turning it at the same rate. This modifies the direction of the force applied by friction. As it is always parallel to the axle, the direction is changed as the car moves forward.
The consequence of this is that if the forward speed is constant, the magnitude of the friction (and therefore the acceleration and torque) are also constant.
A force/acceleration that is perpendicular to the direction of motion (which is what an ideal wheel does) will create circular motion.
It seems like you are looking for some part of this to be connected, but I'm not sure what part that is.
- The wheel generates a force (due to friction) parallel to the axle.
- The direction of the force is determined by the orientation of the wheel.
- The wheel's orientation is fixed (for this problem) with the body of the car.
- The car turns because of torques from the front and rear wheels.
- A force that is applied perpendicular to motion creates (in certain cases) circular motion.