Immersed boundary (IB) methods have emerged as an attractive option to represent a stationary or moving body in fluid flow simulations. For incompressible flows, which is the focus of the present work, most IB implementations have been done in the framework of projection methods and in many cases significant errors in satisfying the impermeability constraint on a solid body have been reported. Such errors, however, strongly depend on the specifics of the formulation, as well as the flow configuration. Various adaptations of IB methods have been proposed to tackle this issue, which either construct modified divergence operators, or treat the pressure field as an additional Lagrange multiplier replacing the standard Poisson equation by a more complex system. The latter comes at the additional cost of solving a non-standard coupled Poisson problem. In this work, we propose a new formulation to enforce appropriate boundary conditions on the pressure on the IB as part of the solution of the Poisson equation in a fractional step approach. The overall treatment is inspired by the ghost fluid method (GFM). The main advantage of our approach is that a standard, constant coefficient Poisson equation is solved, and all modifications to enforce the boundary conditions are introduced in the RHS. As a result, standard fast solvers based on trigonometric transformations can be utilized. The accuracy and robusteness of the proposed formulation will be demonstrated in variety of flow problems of increasing complexity. The case of a very thin solid object between two channels with flows in the opposite directions, for example, produces large errors in the pressure evolution when computed with standard IB methods. These errors are eliminated with the proposed formulation. Several other challenging external (flow around a sphere and a very thin airfoil) and internal (model left ventricle of the heart) flow problems will be presented.