In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for parallel to pi u - u(h)parallel to(E), where pi u is some interpolant of the solution u and u(h) the discrete solution. This supercloseness result implies an optimal error estimate with respect to the L(2) norm and opens the door to the application of postprocessing for improving the discrete solution.