In this work, we present an immersed boundary-simplified gas kinetic scheme for simulation of two-dimensional (2D) incompressible flows with curved and moving boundaries. Specifically, a fractional step technique with predictor and corrector processes is introduced to solve the governing equations. In the predictor step, the macroscopic governing differential equations are solved on the fixed Eulerian meshes by the recently developed simplified gas kinetic scheme (GKS). Compared to the conventional GKS, the simplified GKS is simpler and more efficient. At the same time, the simplified GKS inherits the advantage of good robustness of conventional GKS. In the corrector step, the velocity correction is carried out on the Lagrangian points by the implicit boundary condition-enforced immersed boundary method (IBM). Since it strictly originates from the no-slip boundary condition, this approach can avoid completely the unphysical streamline penetration phenomenon. Several numerical experiments show that the 2D incompressible flows with curved and moving boundaries can be well simulated by the developed scheme.