It is well known that the minimax rates of convergence of nonparametric density and regression function estimation of a random variable measured with error is much slower than the rate in the error free case. Surprisingly, we show that if one is willing to impose a relatively mild assumption in requiring that the error-prone variable has a compact support, then the results can be greatly improved. We describe new and constructive methods to take full advantage of the compact support assumption via spline-assisted semiparametric methods. We further prove that the new estimator achieves the usual nonparametric rate in estimating both the density and regression functions as if there were no measurement error. The proof involves linear and bilinear operator theories, semiparametric theory, asymptotic analysis regarding Bsplines, as well as integral equation treatments. The performance of the new methods is demonstrated through several simulations and a data example.