This work presents a nonlinear error estimation and correction approach for the Reynolds-averaged Navier-Stokes (RANS) equations, coupled with a norm-oriented (NO) error estimate for anisotropic mesh adaptation. Anisotropic mesh adaptation aims at reducing the interpolation error of the numerical solution and relies on an error estimate to compute the metric and generate the mesh. The choice of the error estimate has a significant impact on the efficiency of the mesh adaptation process. Goal-oriented (GO) error estimates are widely used in aeronautical application, as they adapt the mesh with respect to a specific scalar quantity of interest, such as drag or lift. However, due to their formulation, GO approaches may neglect important flow features that do not directly influence the chosen scalar output. The norm-oriented approach extends the GO formulation to the entire solution field. By replacing the scalar output with the discretization error itself, the resulting error estimate directly improves the convergence of the solution field. Nevertheless, this approach requires knowledge of the exact solution, which is generally unavailable. To overcome this limitation, we propose a nonlinear corrector formulation for the RANS equations. Using local mesh refinement and flux computation, a corrected solution is obtained from a mesh whose size is locally divided by a factor of two with respect to the current mesh. This corrected solution is then used as a surrogate exact solution in the NO formulation, allowing the computation of the error estimate. The proposed method will be validated on several aeronautical test cases. The results will show that the nonlinear corrector significantly reduces the discretization error. Furthermore, we will demonstrate that the NO error estimate improves mesh convergence with respect to the full solution field and is more efficient than GO or feature-based error estimates.