The modern steer-by-wire (SBW) systems represent a revolutionary departure from traditional automotive designs, replacing mechanical linkages with electronic control mechanisms. However, the integration of such cutting-edge technologies is not without its challenges, and one critical aspect that demands thorough consideration is the presence of nonlinear dynamics and communication network time delays. Therefore, to handle the tracking error caused by the challenge of time delays and to overcome the parameter uncertainties and external perturbations, a robust fast finite-time composite controller (FFTCC) is proposed for improving the performance and safety of the SBW systems in the present article. By lumping the uncertainties, parameter variations, and exterior disturbance with input and output time delays as the generalized state, a scaling finite-time extended state observer (SFTESO) is constructed with a scaling gain for quickly estimating the unmeasured velocity and the generalized disturbances within a finite time. With the aid of the SFTESO, the robust FFTCC with the scaling gain is designed not only for ensuring finite-time convergence and strong robustness against time delays and disturbances but also for improving the speed of the convergence as a main novelty. Based on the Lyapunov theorem, the closed-loop stability of the overall SBW system is proven as a global uniform finite-time. Through examination across three specific scenarios, a comprehensive evaluation is aimed to assess the efficiency of the suggested controller strategy, compared with active disturbance rejection control (ADRC) and scaling ADRC (SADRC) methods across these three distinct driving scenarios. The simulated results have confirmed the merits of the proposed control in terms of a fast-tracking rate, small tracking error, and strong system robustness.