The 4H-silicon carbide on insulator (4H-SiCOI) has recently emerged as an attractive material platform for integrated photonics due to its excellent quantum and nonlinear optical properties. Here, we experimentally realize one-dimensional photonic crystal nanobeam cavities on the ion-cutting 4H-SiCOI platform. The cavities exhibit quality factors up to 6.1×103 and mode volumes down to 0.63 × (λ/n)³ in the visible and near-infrared wavelength range. Moreover, by changing the excitation laser power, the fundamental transverse-electric mode can be dynamically tuned by 0.6 nm with a tuning rate of 33.5 pm/mW. The demonstrated devices offer the promise of an appealing microcavity system for interfacing the optically addressable spin defects in 4H-SiC.