A postprocessing-based a posteriori error estimates for the spectral Galerkin approximation of one-dimensional fourth-order boundary value problems are developed. Our approach begins by introducing a novel postprocessing technique aimed at enhancing the accuracy of the spectral Galerkin approximation. We prove that this postprocessing step improves the convergence rate in both $L^2$- and $H^2$-norm. Using postprocessed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact as the polynomial degree increases. We further extend the postprocessing technique and error estimators to more general one-dimensional even-order equations and to multidimensional fourth-order equations. The results of numerical experiments illustrate the efficiency of the error estimators.