The development of intelligent driving and X-by-wire chassis technology increases the computational complexity and the number of CAN bus-mounted nodes of steering-by-wire vehicles, which leads to non-negligible large random time delay (TD) of the SbW system's control input. TD has a great impact on SbW system's tracking error, which will inevitably deteriorate the safety and yaw stability of vehicles and even cause sideslip accidents. In order to ensure the safety of SbW vehicles, a layered time-delay robust control strategy (LTDRCS) consisting an upper and a lower controller is proposed. Specifically, a novel Lyapunov-Krasovskii (L-K) TD H∞ controller (H∞C) is designed as the lower controller to reduce tracking error, and the system's stability and convergence are synchronously guaranteed by an L-K function and an H∞ norm constraint. As for the upper controller, a novel terminal sliding mode controller (NTSMC) is established to control the yaw rate index and sideslip angle index. In order to restrain the serious influence of the SbW system's inevitable, unknown, and bounded tracking error caused by TD on vehicle yaw stability and ensure the rapid convergence of the system, a novel integral term is introduced. Simulation and experiments show that the proposed strategy efficiently improves the tracking accuracy of the SbW system and the vehicle yaw stability under random TD condition, which benefits from the synergy of the upper and lower layers.