We present a dynamic decomposition analysis of the wake flow in fluid-structure interaction (FSI) systems under both laminar and turbulent flow conditions. Of particular interest is to provide the significance of low-dimensional wake flow features and their interaction dynamics to sustain the free vibration of a square cylinder at a relatively low mass ratio. To obtain the high-dimensional data, we employ a body-conforming variational fluid-structure interaction solver based on the recently developed partitioned iterative scheme and the dynamic subgrid-scale turbulence model for a moderate Reynolds number (Re). The snapshot data from high-dimensional FSI simulations are projected to a low-dimensional subspace using the proper orthogonal decomposition (POD). We utilize each corresponding POD mode for detecting features of the organized motions namely the vortex street, the shear layer and the near-wake bubble. We find that the vortex shedding modes contribute solely to the lift force, while the near-wake and shear layer modes play a dominating role to the drag force. We further examine the fundamental mechanism of this dynamical behavior and propose a force decomposition technique via low-dimensional approximation. To elucidate the frequency lock-in, we systematically analyze the decomposed modes and their dynamical contributions to the force fluctuations for a range of reduced velocity at low Re laminar flow. We ascertain quantitatively that the shear layer feeds the vorticity flux to the wake vortices and the near-wake bubble during the wake-body synchronization. Based on the decomposition of wake dynamics, we suggest an interaction cycle for the frequency lock-in during the wake-body interaction, which provides the inter-relationship between the high amplitude motion and the dominating wake features. Through our investigation of wake-body synchronization below critical Re range, we discover that the bluff body can undergo a synchronized high-amplitude vibration due to flexibility-induced unsteadiness. Owing to the wake turbulence at a moderate Reynolds number of Re = 22, 000, a distorted set of POD modes and the broadband energy distribution are observed, while the interaction cycle for the wake-synchronization is found to be valid for the turbulent wake flow.