Why do you lean towards the left when your car turns right?
Because of inertia.
Newton's 1st Law basically says that objects tend to continue their previous state of motion, unless acted on by an outside force. This is the behavior of inertia.
Assume you are in a country where the driver sits on the right and you are sitting in the left seat. So if the car was traveling North, but turns to go East, your body wants to continue going North. You are sitting in a bucket seat that forces your bottom to change direction to the East. But your upper torso prefers to continue going North. Because after all, that is the Law. I paraphrased it in the first paragraph above.
But the car has changed to going East, so your shoulder collides with the door which gives you a push to the East.
I hope this helps,