We report the timing and spectral analyses of the type-II X-ray bursts from the rapid burster (MXB 1730-335) observed by the Hard X-ray Modulation Telescope (Insight-HXMT) and Swift/X-Ray Telescope (XRT). By stacking the long-duration bursts, we find for the first time that the hard X-rays are lagging behind the soft X-rays by 3 s. However, such a lag is not visible for the short-duration bursts, probably because of the poor statistics. For all bursts the energy spectrum is found to be nonthermal, thanks to the broadband coverage of Insight-HXMT. These findings provide new insights into the type-II bursts and require a temporally visible corona for possible interpretation.