The nonlinear Hall effect (NLHE) holds significant promise for next-generation electronic device applications, and it is essential to explore and engineer a strong and highly tunable NLHE. In this work, we report large second-order and third-order NLHEs in dual-gated BiSbTeSe2 devices. Both the second-order and third-order NLHE are highly gate-tunable. The second-order NLHE exhibits a higher generation efficiency than those reported in previous topological insulator systems, and it can be stably observed up to room temperature. Meanwhile, the generation efficiency of the third-order NLHE is comparable to previously reported values. The scaling analysis indicates that the second-order NLHE is consistent with contributions from skew scattering and a term compatible with either the Berry curvature dipole or side jump, whereas the third-order NLHE is consistent with contributions from the Berry connection polarizability tensor and and higher-order skew scattering. Our work establishes a promising experimental platform for the development of NLHE-based devices.