First we need to understand the mechanism and free body diagram of the horse cart system.

When the horse is tied with the cart , he pushes the ground with his feet in backward direction and due to friction the horse gets reaction and it moves in forward direction , as the horse moves in forward direction he pulls the cart with him , when the horse pulls the cart he does action and the cart pulls him towards it and that is reaction but the horse adjusts it’s muscular power to not to get pulled by cart but to pull the cart. The friction acting on the wheels of the cart makes the wheels roll and hence the cart moves.

Note that the action reaction pair of weight and normal force are canceled out because they are equal and opposite by Newton’s third law.

Actually the simple reason is that the horse moves with suitable force in order to pull the cart with it and not to struggle by applying less force. The main action reaction pair is between that of horse and cart due to the rope attaching them both. The tension in the rope is exerted on the cart in rightward direction and on the horse the tension is exerted in the leftward direction along with the friction force which acts in rightward direction.

To understand this completely we should simplify our system by considering cart at one time and horse at another time independently

CART ALONE – The only driving force on the cart is the tension which acts rightward or forward , hence the net force is also tension , so by Newton’s second law through the ropes the cart moves.

HORSE ALONE – There are two forces acting on the horse which are not cancelled by each other the one is tension in the ropes which pull the horse backwards and the friction force from the ground which makes the horse moves forward.