Competitive propagation of fractures initiated from multiple perforation clusters is universal in hydraulic fracturing of unconventional reservoirs, which largely influences stimulation. However, the propagation mechanism of multi-fractures has not been fully revealed for the lack of a targeted laboratory observation. In this study, a physical simulation experiment system was developed for investigating the initiation and propagation of multi-cluster hydraulic fractures. Different from the traditional hydro-fracking test system, the new one was equipped with a multi-channel shunting module and a strain monitoring system, which could guarantee the full fracture extension at each perforation clusters and measure the internal deformation of specimens, respectively. Several groups of true tri-axial fracturing tests were performed, considering the factors of in situ stress, cluster spacing, pumping rate, and bedding structures. The results showed that initiation of multi-cluster hydraulic fractures within one stage could be simultaneous or successive according to the difference of the breakdown pressure and fracturing fluid injection. For simultaneous initiation, the breakdown pressure of the subsequent fracture was lower than or equal to the value of the previous fracture. Multiple fractures tended to attract and merge. For successive initiation, the breakdown pressures of fractures were gradually increasing. The subsequent fracture tended to intersect with or deviated from the previous fracture. Multiple fractures interaction was aggravated by the decrease of horizontal stress difference, bedding number and cluster spacing, and weakened by the increase of pump rate. The propagation area of multiple fractures increased with the pump rate, decreased with the cluster spacing. The strain response characteristics corresponded with the initiation and propagation of fracture, which was conducive to understanding the process of the fracturing. The test results provide a basis for optimum design of hydraulic fracturing.