We studied the nonlinear electric response in WTe₂ and MoTe₂ monolayers. When the inversion symmetry is breaking but the the time-reversal symmetry is preserved, a second-order Hall effect called the nonlinear anomalous Hall effect (NLAHE) emerges owing to the nonzero Berry curvature on the nonequilibrium Fermi surface. We reveal a strong NLAHE with a Hall-voltage that is quadratic with respect to the longitudinal current. The optimal current direction is normal to the mirror plane in these two-dimensional (2D) materials. The NLAHE can be sensitively tuned by an out-of-plane electric field, which induces a transition from a topological insulator to a normal insulator. Crossing the critical transition point, the magnitude of the NLAHE increases, and its sign is reversed. Our work paves the way to discover exotic nonlinear phenomena in inversion-symmetry-breaking 2D materials.