Based on the idea of classical discontinuous Galerkin and weak Galerkin finite element methods, we introduce an over-penalized term in the numerical schemes as a part of stabilization for solving PDEs. From the double-valued functions defined on interior edges of elements, it is natural to generate jumps of the over-penalized term. An over-penalized weak Galerkin (OPWG) finite element method can be applied well to several interface problems with general imperfect interface. Optimal error estimates of those schemes in the $L^2$ and energy norms can be proved.