I try to stand on water surface but I just drown. A book resting on the table because of the W(gravitational Force) and normal force of the table acting on the book oppositely in equal magnitude being canceled out. When it comes to a fluid (air, water, oil etc) If I put my two feet on the surface I will simply drown. Why that? See, the weight of my body is acting on the surface and according to the third law at that instant there will be an equal and opposite force acting on me from the surface. So should not I be able to stand? If you are about to say that fluid molecules have surface tension and they are not steady then I would like to say the little portion of weight of any point of my body that pushes the tiny fluid molecule, that tiny part should cancel that little amount of weight. By this each and every point of my body (along with my Centre of gravity) part should have a net 0 force which should make stand on the fluid surface theoretically. But in practical case it doesn't happen. Why that? I can stand on frozen Lake but can't stand when it is not frozen.
Just to start, your initial statement of:
I try to stand on water surface but I just drown.
Isn't completely accurate. As long as you have some air in your lungs, it's possible to stay afloat; because the human body is less dense than water, and floats due to buoyancy. By positioning your head correctly, it will be above the water and you would be able to breathe.
Now, to address your main question about Netwons Third law, I believe you are just mixing up what forces to consider. It's true that every action will have an equal and opposite reaction. That doesn't mean that all the weight of the object is exerted on the fluid.
In the case of an object falling into a pool of liquid, there is an opposite and equal action from the water; but this isn't necessarily equal to the total weight of the object. For a sinking object, it is initially less than the weight of the object. This means there is a net force (and therefore net acceleration) downwards on the sinking body. This does not violate the third law; because part of the force is acting to accelerate the mass, and the rest of the weight has an equal and opposite reaction with the fluid.
Also, with enough viscous drag, eventually the force from the fluid will be equal to the weight of the object; but due to Newtons First Law, we know if the forces are balanced, it will keep moving at the same speed and sink, but it will not keep accelerating down.
This is because liquids can't resist shear stress. Your body makes contact with a part of the surface of the liquid and just outside the point in lateral direction which is not under your feet feels a shear force. Due to this liquid surface moves and you keep on pushing down until all of the liquid has been displaced (assuming you are heavy enough).
What you can do is that you can try to make the contact with whole of the liquid surface, then there won't be any such force. To do so you can lay down a plank on the surface covering whole of it then you can stand on it. In this case liquid will support your weight.
You exert a force mg on water, so water exerts mg back on you. So now you're in equilibrium. But the water has net downward force so it moves . Now there is no reaction force on you as you're no longer in contact with water, so you experience a net downward force so you also move downward. This continues until water cannot move. This is the reason we can't stay in equilibrium on similar things which can be solid, liquid,gas...