If an object moving in a circle experiences centripetal force, then doesn't it also experience centrifugal force, because of Newton's third law?

This is a common misinterpretation of Newton's third law, often stated as "to every action, there's an equal and opposite reaction." As you surmise, "action" and "reaction" refer to forces. However, they refer to forces acting on different things. Otherwise, nothing could accelerate, ever: if every force were always canceled out by an equal and opposite force, no force could ever do anything. Instead, forces occur between objects--say car and road, to take your example. The road exerts an inward force on the car, which, you're right, is the centripetal force. The equal and opposite force is exerted by the car, on the road. The two forces are acting on different things, so they do not cancel. This second force (the force exerted by the car on the road) is sometimes referred to as the "reactive centrifugal force," which is confusing, because it's different from the more common meaning of centrifugal force.

Lets look at the Earth-moon system for an example. The centripetal force is Earth's gravity, keeping the Moon from flying away. But this works both ways, the Earth is pulled towards the Moon just as hard as the moon is pulled towards the Earth.

In your car example, the angle of the front tires means some percentage of the force of the car is spent on turning the car. The opposite force is spent trying to push the roadway in the opposite direction. It's the same as driving forwards really, except your force vector isn't parallel with your velocity vector.

Quick little aside: Newton's laws, the ones you learn in High-school anyways, only work in inertial reference frames. Centrifugal force does exist in a rotating reference frame.

Imagine an object connected by a string moving in a circular motion.

what is actually this reaction force that's created by the centripetal force?

The force on a object, which causes the centripetal acceleration of an object, is due to another entity - the action, eg the force on the object due to the string.

The Newton third law pair is the force on another entity due to the object - the reaction, eg the force on the string due to the object.

where does the centrifugal force come from?

The centrifugal force is not a real force, rather it is introduced for the convenience of being able to use Newton’s second law in the rotational (non-inertial) frame of the object.

There is no Newton third law pair to the centrifugal force.