I normally find that I can explain equal and opposite forces not having the same result in Newton's third law because those forces are acting on objects with different masses. However, this doesn't work for the following situation, and I haven't managed to resolve the seeming paradox in it yet.
Say you had a horse of 1 kg, pulling a cart of 1 kg. The horse exerts 5 N forward on the cart, and the cart exerts 5 N backward on the horse (I would draw a diagram, but I don't know how to with this software: sorry!). The horse is succeeding in pulling the cart forwards. If the horse is accelerating forwards with the cart, then there must be a resultant forward force on the horse. If you said that the horse was pushing on the ground where the horizontal component of his force was 10 N backwards, meaning the horizontal component of the ground's force on the horse is also 10 N forwards, then you could argue that, with 5 N backwards from the cart and 10 N forward from the ground, there is a resultant force of 5 N forwards on the horse. However, the horse can only generate the pulling force of 5 N on the cart by using this forward force that the ground is exerting on him/her, so surely this means that both of these forward forces must be equal? In which case, doesn't this mean that the ground could only be exerting 5 N forward on the horse if the horse was only managing to exert 5 N forward on the cart, meaning that when you factor in the 5 N backwards that the cart is exerting on the horse, the resultant force on the horse is 0, and it is impossible for the horse to accelerate the cart? And since it isn't impossible, what have I not factored in/reasoned wrongly here?