In this paper, the problem of controller design for a class of switched nonlinear time-delay systems with finitetime output constraints is studied. In order to limit the output tracking error to a predetermined boundary in finite time, an improved performance function, finite time performance function (FTPF), is introduced. In the process of controller design, barrier Lyapunov function is used to solve the problem of output constraint. Then, a new adaptive MTN control method is proposed by combining the universal approximation property of multi-dimensional Taylor network (MTN) with adaptive backstepping technology. In this method, Lyapunov-Krasovskii functionals are designed to ensure the stability of the switched system in the presence of time-delay. Finally, two simulation examples are given to illustrate the effectiveness and practicability of the design scheme.