We study multi-follower bilevel optimization problems with binary linking variables where the second level consists of many independent integer-constrained subproblems. This problem class not only generalizes many classical interdiction problems but also arises naturally in many network design problems where the second-level subproblems involve complex routing decisions of the actors involved. We propose a novel branch-and-cut decomposition method that starts by solving the first level and then iteratively generates second-level feasibility and optimality cuts that are obtained by solving a slightly adjusted second-level problem. Compared to many other existing solution methods, we do not rely on solving the High Point Relaxation of the bilevel problem but fully project out the second level, resulting in significant computational advantages when the second-level problem is very large or possesses a weak LP relaxation. Also, our approach can be fully automated within a MIP solver, making it very easy to apply for those who do not want to design problem-tailored algorithms for their bilevel problem. Computational results for a bilevel network design problem demonstrate that our approach efficiently solves instances with hundreds of subproblems in a few minutes, significantly outperforming the Benders-like decomposition from the literature on challenging instances.