This work presents a versatile and robust immersed edge-based smoothed finite element method with the mass conservation algorithm (IESFEM/Mass) to solve partitioned fluid-structure interaction (FSI). A gradient smoothing technique was used to solve the system governing equations, which can improve the calculated capability of the linear triangular elements in two phases. Based on the quadratic sharp interface representation of immersed boundary, an extended fictitious domain constructed by a least squares method approximately corrected the residual flux error. The compatibility for boundary conditions on moving interfaces was satisfied, thus eliminating spurious oscillations. The results from all numerical examples were consistent with those from the existing experiments and published numerical solutions. Furthermore, the present divergence-free vector field had a faster-converged rate in the flow velocity, pressure, and FSI force. Even if in distorted meshes, the proposed algorithm maintained a stable accuracy improvement. The aerodynamics of one- and two-winged flapping motions in insert flight has been investigated through the IESFEM/Mass. It can be seen that the wing-wake interaction mechanism is a vital factor affecting the lift. The applicability of the present method in the biological FSI scenario was also well-demonstrated.