This article investigates the problem of global stabilisation for a class of switched nonlinear systems with unknown control coefficients by output feedback. Full state measurements are unavailable. We first show that via a coordinate transformation, the unknown control coefficients are lumped together and the original switched nonlinear system is transformed into a new switched nonlinear system for which control design becomes feasible. Second, for the new switched nonlinear system, based on backstepping, we design output-feedback controllers for subsystems and construct a common Lyapunov function, which rely on the designed state observer, to guarantee asymptotic stability of the closed-loop system under arbitrary switchings. Finally, as an application of the proposed design method, global stabilisation of a mass-spring-damper system is achieved by output feedback.