Summary: This paper is concerned with the problem of delay and mode-dependent robust \(\mathcal{H}_\infty\) nonrational Dynamic Output Feedback Controller (DOFC) synthesis for a class of continuous-time Semi-Markovian Jump Linear Systems (S-MJLSs) with time-varying delay. Due to the relaxed conditions on the stochastic process, the S-MJLSs are with time-varying transition rates and can describe a broader class of dynamical systems than the traditional Markovian jump linear systems. By introducing a two-term approximation for the time-varying delay, the original system is firstly reformulated into a feedback interconnection configuration, which is well-posed in the sense that the Scaled Small Gain (SSG) technique can be applied to the reformulated system to derive robust performance analysis criteria. Then, based on a semi-Markovian Lyapunov-Krasovskii formulation of SSG condition combined with the sojourn-time fractionizing technique, the \(\mathcal{H}_\infty\) performance analysis and mode-dependent nonrational DOFC synthesis conditions for the underlying S-MJLSs are developed, respectively. It is shown that the controller gains can be obtained in terms of linear matrix inequalities. Finally, simulation studies are provided to illustrate the effectiveness and superiority of the proposed design method.