The vehicle turns because of applied Lateral tire forces . While turning the tire experience a centrifugal or inertial force, which lead to the deformation of tire tread at bottom contact patch and their is the generation of forces and moments.
Let's first see how lateral force is generated ?
In the above figure , you can see that when you take a right turn then the car tire experience a centrifugal or inertial force at wheel center. This inertial force Fc , tries to pull the vehicle towards left direction and to prevent this pulling a force is generated at bottom of tire in right direction , which tries to resist this pulling . The resisting force here , is shown by Fy. Fy is called the lateral force and the maximum value of Fy is given by :-
Fy = μy * W
Where μy is the coefficient of friction in y direction.
Now let's see this case at micro level:-
Here , tire treads are shown by brush bristles. This also called the tire brush model. Now when the tire is pushed leftwards due to centrifugal force Fc , the treads on the ground get stretched and generate a tension in them. This tension developed in the tire treads is called Lateral force , shown by Fy .
When Fc > Fy , the tire will skid towards left direction.
This is the bottom contact patch of tire while turning . When the tire takes a right turn, it experience a lateral force Fc towards right direction and a longitudinal force Fx in forward direction.
Due to these forces Fx and Fc , tires contact patch deforms like a curve, shown in figure.
Let's see why this is happening,
Suppose , you are a black dot shown in the above figure, now currently you are in un-deformed region of tire. Now as the tire rolls , you hit the contact patch at first black dot position. At this point you are acted by the forward force Fx and the lateral force Fy in right direction , due to which , you will be moving in black arrow direction. This will stretch you away from the central line C2 of tire and you will move to second position of black dot , then to third and so on.
You will be stretched till the maximum value of Fy is achieved , which is generally achieved after some distance from the central line C1 . When the maximum value of Fy is achieved , now beyond this point as the tire tread is unloaded , you will get skid and move towards the plane C1 of tire and this cycles continues . This is happening with every tread of tire , due to which the tire is deformed like a curve at contact patch.
Tangent made to the tires contact patch (green arrow) , shows the actual moving direction of tire. Angle between this tangent and the direction in which tire is pointing is called Slip angle and is denoted by α .
Lateral force Fy is acting at some distance "a" from the central line C2 , and this distance is called pneumatic trail.
Now here, Fy is the lateral force generated at some distance "a" from the center. This force will produce a self aligning torque at center and will try to rotate the tire clockwise direction.
Mz - self aligning torque
When slip angle & lateral force is reduced,it will reduce the pneumatic trail "a" and will reduce the self aligning torque.