In recent years, swimming robots have been developed to achieve efficient propulsion and high maneuverability that are possessed naturally by fish. Previous studies have attempted to achieve swimming similar to fish by control based on physical models and top-down architectures, but have encountered problems due to the high complexity of the underwater environment. Several research works have tried to overcome these problems by exploiting embodiment—that is, by mimicking the physical properties of fish. To achieve more intelligent swimming from the perspective of the embodiment, we focused on a framework called physical reservoir computing (PRC). This framework allows us to utilize physical dynamics as a computational resource. In this study, we propose a soft sheet-like swimming robot and a PRC-based architecture that can be used to emulate swimming motions by exploiting its own body dynamics for closed-loop control. Through experiments, we demonstrated that our system satisfies the properties required for learning swimming motion through supervised learning. We also succeeded in robust motion generation and environmental state estimation, opening up future prospects for more intelligent robot control and sensing.