In this work, we introduce a new nonlinear stabilization approach for high-order continuous finite element discretizations of scalar conservation laws. The proposed methodology is based on the weighted essentially non-oscillatory (WENO) framework. Unlike Runge-Kutta discontinuous Galerkin (RKDG) schemes that overwrite the finite element solution with a WENO reconstruction, our scheme uses a reconstruction-based smoothness sensor to blend the numerical viscosity coefficients of high- and low-order dissipative stabilization terms. The so-defined WENO approximation introduces low-order nonlinear diffusion in the vicinity of shocks and retains high-order accuracy of a continuous Galerkin scheme with linear stabilization in regions where the solution is sufficiently smooth. The reconstructions that we use include Lagrange and Hermite interpolation polynomials. The amount of numerical viscosity depends on the differences between partial derivatives of candidate polynomials. All derivatives are taken into account by our smoothness sensor. The preliminary results for standard linear and nonlinear test problems in one and two dimensions are promising. In our numerical experiments, we observe crisp resolution of shocks and optimal convergence behavior even for high polynomial degrees.