In this paper, a second-order linearized discontinuous Galerkin method on general meshes, which treats the backward differentiation formula of order two (BDF2) and Crank–Nicolson schemes as special cases, is proposed for solving the two-dimensional Ginzburg–Landau equations with cubic nonlinearity. By utilizing the discontinuous Galerkin inverse inequality and the mathematical induction method, the unconditionally optimal error estimate in $L^2$-norm is obtained. The core of the analysis in this paper resides in the classification and discussion of the relationship between the temporal step size $\tau$ and the spatial step size $h$, specifically distinguishing between the two scenarios of ${\tau }^2\le h^{k+1}$ and ${\tau }^2>h^{k+1}$, where $k$ denotes the degree of the discrete spatial scheme. Finally, this paper presents two numerical examples involving various grids and polynomial degrees to verify the correctness of the theoretical results.