Numerical modeling of multiphase flows at high Reynolds numbers is a challenging task. Traditional methods based on sharp-interface models might encounter numerical difficulty in handling rapid topological changes such as breakup and coalescence. Therefore, diffuse-interface models are widely used for numerical simulation of multiphase flows. As a diffuse-interface model, the lattice Boltzmann method (LBM) is a well-established mesoscopic scheme for numerical study of complex fluids. The most interesting feature of the LBM is that all nonlinearity is local and all nonlocality is linear, which is favorable for utilization on massively parallel machines. In this study, we increase the numerical stability of the LBM for multiphase flows at high Reynolds numbers, and investigate the Kelvin-Helmholtz instability of a shear flow. In order to save the computational resources, we propose an efficient adaptive mesh refinement (AMR) algorithm which does not need to maintain or modify a tree data structure. We then reformulate the LBM on nonuniform grids and propose an AMR-LBM for two-phase flows. Various case studies such as rising bubble and falling drop under buoyancy force, drop splashing on a wet surface, and droplet coalescence onto a fluid bath are conducted for validation and verification. The Kelvin-Helmholtz instability of a stratified shear-layer flow is also scrutinized to assess the accuracy of the proposed model, and satisfactory agreement with benchmark studies is shown.