The high-order mode oscillation is studied in designing a four-cavity intense relativistic klystron amplifier(IRKA). The reason of the oscillation caused by highorder modes and a method to suppress this kind of spurious modes are found through theoretical analyses and the study on the influence of major parameters of a high frequency structure (such as, the oscillation frequency of cavities, the cavity Q value, the length of drift tube section and the characteristic impedance etc.). Based on much simulation, a four-cavity Intense Relativistic Klystron Amplifier with a superior performance has been designed, built and tested. An output power of 2.22 GW corresponding to 27.4% efficiency and 61 dB gain has been obtained. Moreover, the high-order mode oscillation is suppressed effectively, And an output power of 1.95 GW corresponding to 26% efficiency and 62 dB gain has been obtained in the laboratory.