Question

if action is always equal to the reaction then how a horse is able to move a cart

Answer 1

              Action is equal and opposite to the reaction. Yes. Even in the case of a horse pulling the cart. The horse can pull the cart forward and walk if and only if the ground it is walking on pushes its hoof with a force. That force is friction.
In the absence of that reaction force called friction, it would be like walking on a smooth ice floor

Answer 2

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When the horse walks in the forward direction (with the cart attached to it), it exerts a force in the backward direction on the Earth. An equal force in the opposite direction (forward direction) is applied on the horse by the Earth. This force moves the horse and the cart forward.

The velocity at which the horse can move by applying a force on the earth depends on the mass of the horse (and the cart attached to it). The heavier the cart, the slower the motion of the horse (for a given amount of force applied by the horse on the Earth). If the cart is too heavy, the force exerted by the horse on the Earth will be insufficient to even overcome the force of inertia. In this case, the horse will not be able to pull the cart.