When we walk we push the Earth backwards and in return the Earth exerts an equal and opposite force on us in the form of static friction acting in the same direction as we walk. But what if we slip (backward) while walking?Doesn't it mean that we exert a greater force on the ground than that of the Earth on us?
In the case of slipping, we still exert exactly the same amount of force on the ground as the ground exerts back on us. The difference is that we intended to use more force than we could actually apply. Some of that "missing" force goes into accelerating our leg. Some of it is simply an illusion: the human brain does not always perceive exactly how much mechanical force it is exerting.
A similar analogy may be found in weightlifting. You may be capable of lifting with 60 pounds of force. However, if you lift a stack of books that weighs 10 pounds, doesn't that mean you're exerting 60 pounds against 10 pounds of reactionary force? No. In reality, you are putting far less than your maximum force into those books. You are actually putting close (if not exactly) 10 pounds of force into them.
I think you are confusing forces here. When you put your foot on the ground, and step, the weight of your body is the force acting downwards, pushing on the ground. Satisfying the Newton's 3rd Law, the ground's surface provides an equal force in the opposite direction, the normal force, acting directly upwards.
Now, friction is a force in a horizontal direction. So that would be a totally different 3rd Law pair (if such existed). Now, you are right, however, the Weight-Normal pair does have to do something with friction, and slipping. The maximum available force before slipping occurs is:
$$ f_s = μ_sF_n $$
Now any force greater than that, will cause change in acceleration, in this case, it will simply start accelerating the object in the direction of that force.
Normal force is the one that is being "matched" by the ground (Newton's Third Law). Only one 3rd Law pair. Friction, you could say, is a dependent, "resulting", force that depends on the value of the normal force. So, no greater force being applied here.
On the other hand, yes, you can easily apply a force that will be greater than the static friction supplied by the ground or any other surface.