This article investigates the asynchronous fuzzy control for networked vehicle steering by applying a switching event-triggered mechanism and a fully actuated system approach. The nonlinear vehicle steering tracking system is described by a T-S fuzzy model with a polytope modeling method. A switching event-triggered communication mechanism is presented, where two triggering conditions are activated based on state convergence and divergence, respectively. Inspired by fully actuated system approach, an asynchronous fuzzy controller with open-loop system dynamics is constructed in consideration of the event-triggered mechanism and bounded communication delays. The closed-loop system is modeled by a parameterized nonlinear system with input delay under switching event-triggered inequalities. A nonlinear design result of the fuzzy controller is derived by a switching discontinuous Lyapunov-Krasovskii functional, which removes the assumption in existing results that the bounds of asynchronous constraint on membership functions are available. Through comprehensive comparative analysis, it is numerically shown that the proposed result can achieve less conservatism, better control effect, and larger average release intervals.