The steady-shear viscosity, dynamic viscoelasticity, and normal stress behavior were measured for suspensions flocculated by reversible bridging of polymer with weak affinity for the particle surface. Since the bridges are constantly forming, breaking, and re-forming by thermal energy, the flow becomes Newtonian at very low shear rates. When subjected to shear fields, the polymer bridges are highly extended and this produces shear-thickening flow. Although the extension of flexible coil connecting particles is an intrinsic mechanism, the network structure is essential for shear-thickening flow of suspensions. The appearance of shear thickening is analyzed by percolation theory. The shear-thickening suspensions generate a striking normal stress effect, which gives a measure of elasticity. The scaling analysis of elastic behavior near percolation threshold shows that the critical exponent of the reciprocal of the steady-shear compliance with respect to the difference of the particle concentration from the critical value is 1.7. The elastic network of flexible polymer bridges can be modeled by an isotropic force constant. Therefore, the shear-thickening flow for suspensions flocculated by reversible bridging can be explained by the nonlinear elasticity due to entropy effect of extended bridges.