This paper focuses on the problem of robust $H_{infty}$ control of nonlinear switched systems with parameter uncertainty via the multiple Lyapunov functions (MLFs) approach. The uncertain parameters are assumed to be in a known compact set and are allowed to enter the system nonlinearly. Based on the explicit construction of Lyapunov functions, which avoids solving the Hamilton-Jacobi-Isaacs (HJI) inequalities, sufficient conditions for the solvability of the robust $H_{infty}$ control problem of cascade nonlinear switched systems are derived under some switching signal. Then, the result is extended to solve the robust $H_{infty}$ control problem of nonlinear switched systems in strict feedback form. Finally, the effectiveness of the proposed results is illustrated through a simulation example