This paper is concerned with the problem of global output-feedback control for a class of stochastic nonlinear time-delay systems in the presence of input saturation and unmeasurable states. As a distinctive feature, the growth assumptions imposed on the drift and diffusion terms are proven to be unnecessary, which can be removed through a technical lemma. In addition, by introducing an auxiliary system whose order is the same as the considered system, and using Lyapunov-Krasovskii functional, the adverse effects generated by input saturation and time-varying delay are eliminated. Then, based on state-observer and backstepping recursive design, an output-feedback controller is constructed to render the closed-loop system be globally bounded almost surely. Particularly, instead of converging to an arbitrarily small neighborhood of zero as in related results, the tracking error is ensured to be tuned by the design parameters and an input saturation error in the mean quartic sense. Finally, a stochastic chemical reactor system is established and shown to demonstrate the effectiveness of the proposed scheme.