Exponential stability, L1-gain performance and controller design problems for a class of positive switched systems with time-varying delay are investigated in this paper. First, by constructing an appropriate co-positive type Lyapunov-Krasovskii functional, sufficient conditions for the exponential stability are developed by using the average dwell time approach. Then, the weighted L1-gain performance is investigated for the system considered. Based on the results obtained, an effective method is proposed for the construction of a stabilizing feedback controller with L1-gain property. All the results are formulated as a set of linear matrix inequalities (LMIs) and therefore can be easily implemented. Finally, the theoretical results obtained are demonstrated by a numerical example.