We consider numerical simulation of two-phase flows in porous media using implicit methods. Because of the complex features involving heterogeneous permeability and nonlinear capillary effects, the nonlinear algebraic systems arising from the discretization are very difficult to solve. The traditional Newton method suffers from slow convergence in the form of a long stagnation or sometimes does not converge at all. In this paper, we develop nonlinear preconditioning strategies for the system of two-phase flows discretized by a fully implicit discontinuous Galerkin method. The preconditioners identify and approximately eliminate the local high nonlinearities that cause the Newton method to take small updates. Specifically, we propose two elimination strategies: one is based on exploring the unbalanced nonlinearities of the pressure and the saturation fields, and the other is based on identifying certain elements of the finite element space that have much higher nonlinearities than the rest of the elements. We compare the performance and robustness of the proposed algorithms with an existing single-field elimination approach and the classical inexact Newton method with respect to some physical and numerical parameters. Experiments on three-dimensional porous media applications show that the proposed algorithms are superior to other methods in terms of robustness and parallel efficiency.