“a horse pulls on a cart. by newton’s third law of motion, the cart pulls back on
the horse with a force equal to that exerted by the horse on the cart. the sum of
the forces is zero. therefore it follows that it is not possible for the horse to
accelerate the cart.”
is this way of thinking argument correct? explain fully.

If they are both in space and not on ground, then the above way of thinking is correct. Horse will not be able to accelerate forward.

But here, there are some external forces acting on the horse other than the pull of the cart. That makes the horse accelerate. Cart also gets accelerated.

Horse kicks and pushes the road or ground with his feet backwards. So the ground pushes with an equal force in the forward direction. On the horse, there is a force by the cart, pulling it backwards and a force of friction from the ground and a push forward given by the ground. Pushing forward force by the ground is higher than the sum of others. So horse accelerates.