Bilateral control systems often operate in harsh working environments, where equipment wear and aging factors are unavoidable. When the physical parameters of the system change, the originally designed controller may no longer effectively control the system. This paper focuses on the two-link bilateral control system to conduct research on data-driven dynamic modeling and adaptive control. First, the input-output data of the system is used to establish the data-driven dynamic models for both the master and slave systems using Lagrangian Neural Networks (LNN). Then, considering the inevitable errors between the data-driven model and the actual system structure, a four-channel robust adaptive controller is designed based on the data-driven dynamic model, and the stability of the closed-loop control system is proven using Lyapunov stability theory. Finally, simulations are conducted to investigate scenarios where the master side is subjected to different manipulation forces and the slave side experiences various environmental forces. The simulation results indicate the stability and transparency of the proposed controller, ensuring that the slave system can stably track the motion of the master system and effectively reflect the interaction forces between the slave manipulator and the external environment.