There's a centripetal force pulling you to the left. More specifically it's a force pulling you toward the center of the circle. This circle is based on the turning radius. The sharper the turn, the closer the center would be, and hence it causes a stronger force. A more gradual turn means the center is more distant and there's less of a centripetal force.
At the same time there's also inertia. This force is commonly mistaken as "centrifugal" force which is not a real thing. Inertia is the tendency for a person or an object to keep moving in the same direction at the same velocity. Refer to Newton's 1st Law of Motion. It says "A body at rest tends to stay at rest. A body in motion tends to stay in motion. Both conditions are changed when an outside force is applied". Because there's the tendency to keep moving forward, coupled with the leftward force, it gives the illusion of a force pulling outward away from the center of the circle.
So in short there's a centripetal force pulling you inward to make the turn happen in the first place. At the same time there's also inertia making you want to travel at the same speed and the same direction you were heading before the turn. This explains why you feel an outward pulling force, pulling you to the right, when you make that left turn.