Fracture–cavity carbonate reservoirs account for a considerable proportion of oil and gas resources. Because of the complicated relationships between cavities, fractures and pores in these reservoirs, which are defined as cavity clusters, fracturing technology is employed to enhance their hydrocarbon productivity. However, almost all previous studies have just considered the effect of a single natural cavity or fracture on the propagation of a hydraulic fracture; therefore, the mechanism by which a hydraulic fracture interacts with a cavity cluster needs to be clarified. In this study, cavity clusters with different distributions were accurately prefabricated in synthetically made samples, and large-scale simulation equipment was employed to systematically perform fracturing experiments considering different horizontal differential stress levels. Meanwhile, the hydraulic fracture propagation behaviors were comprehensively analyzed through fracture morphology, fracturing curves, the complexity of the fracture network and acoustic emission monitoring. It was found that a natural fracture with a smaller approach angle is favorable in guiding a hydraulic fracture to a cavity. The fracturing curves were divided into the following four types: frequent fluctuations with “step-like” shapes, great fluctuations with slightly lower closure pressure, fluctuations with obviously lower closure pressure, and little fluctuations with obviously lower closure pressure. And different cavity cluster distributions play a dominant role in the complexity of generated hydraulic fracture networks. In addition, AE energy was used to judge the ease of crossing the cavity. The above findings indicated that for the actual exploration and exploitation of carbonate reservoirs, the geological exploration of different fracture–cavity structures in reservoirs would be required, and targeted fracturing engineering designs need to be carried out for different fracture–cavity carbonate reservoirs.