We analyze the superconvergence property of the Galerkin finite element method (FEM) for elliptic convection-diffusion problems with characteristic layers. This method on Shishkin meshes is known to be almost first-order accurate (up to a logarithmic factor) in the energy norm induced by the bilinear form of the weak formulation, uniformly in the perturbation parameter. In the present paper the method is shown to be almost second-order superconvergent in this energy norm for the difference between the FEM solution and the bilinear interpolant of the exact solution. This supercloseness property is used to improve the accuracy to almost second order by means of a postprocessing procedure. Numerical experiments confirm these results.