This paper is concerned with the global regulation by time-varying output feedback for a class of uncertain nonlinear systems with unmeasurable states dependent time-varying growth and unknown control coefficients. Remarkably, in this paper, the systems allow the heavily time-varying nonlinearities (see Assumption 1), and are considerably different from those in the existing closely related literature. Moreover, the unknown system control coefficients don't necessarily belong to any known finite interval (see Assumption 2), and substantially relax the restrictions on the control coefficients in the existing closely related works. These two aspects make the adaptive method technically more difficult to implement, and consequently a time-varying approach is developed in this paper. Motivated by the recent works, a new time-varying high-gain observer is first designed to rebuild the unmeasurable system states, and in addition to compensate the serious uncertainties/unknowns and time variations in control coefficients and system nonlinearities. Then, a time-varying (nonadaptive) output-feedback controller is successfully constructed by using a backstepping approach together with the delicate choice of design parameters, such that all closed-loop system states are globally bounded and ultimately converge to zero.