No, the magnitude of the force exerted by the string on block 1 is not greater than the weight of block 2.

One important thing to know is that the tension T in the string is the same on both sides of the pulley, since the pulley is massless and frictionless. This tension is equal to the magnitudes T₁ and T₂ of the forces exerted by the string on blocks 1 and 2, respectively. Thus, T = T₁ = T₂.

Now, if T were greater than the weight W₂ of block 2, the net force on this block would be directed upward and, by Newton's second law, block 2 would be accelerating upward. That is not what is happening.