When a car turns does any part of it cross to the other path? I always believe when a car is making turn, some parts of it (either the front corner or the back overhang) will go more into my path momentarily, is this true? Does it make a difference if the car is doing a sharper turn vs. a bigger turn?
I am not talking about skidding, just imaging a car going constant speed forward and turns 90 degree to the left. As the car goes forward, we draw a straight parallel line to its body touching its right side (ignores the side mirror). Now the car turns, would any part cross this line briefly?
 A: It depends on the design of the car.
In a well designed one, it wouldn't happen (top diagram).  The designers would have considered the minimum radius of curvature for the circle that the car can make, and the distance of the back of the car behind the back wheels.
It may even be law that it can't happen (possibly depending on the country where it's sold).

However in principle it could happen (lower diagram), the purple overhang is too far behind the back wheels and could cross the red line.
A: A car turns about its rear wheels, and thus the front swings wide on tight turns.
Specifically, if you track the front of the car (red curve) and the back of the car (blue curve) you will see the front swings wider with a turn radius of $d$

If the turn radius is $r$ (measured from the back wheels) and the length of the car (wheel track) is $L$ then the front axle swings with a radius of $$ d = \sqrt{L^2+r^2}$$
A: A car turns with its front wheels only, the back wheels just follow along. Because of this, the long axis of a turning car is always at an angle to the direction of travel, unlike a car driving straight, which have the axis of the car and the direction of travel perfectly aligned. Because of this, the front corner of a turning car overhangs the usual width of the car, and takes up more width on the outside of the curve than a car going straight would.
The description in the question is a little vague, it's not clear to me how a car could overhang a region defined by the edge of the car itself. That said, I think you may be describing the green region in the diagram below, which represents the front corner overhang that results from the car turning.

Image found here.
