Research on network percolation and synchronization has deepened our understanding of abrupt changes in the macroscopic properties of complex engineered and natural systems. While explosive percolation emerges from localized structural perturbations that delay the formation of a connected component, explosive synchronization is usually studied by fine-tuning of global parameters. Here, we introduce the concept of synchronization bombs as large networks of heterogeneous oscillators that abruptly transit from incoherence to phase-locking (or vice-versa) by adding (or removing) one or a few links. We build these bombs by optimizing global synchrony with decentralized information in a competitive percolation process driven by a local rule, and show their occurrence in systems of Kuramoto –periodic– and Rössler –chaotic– oscillators and in a model of cardiac pacemaker cells, providing an analytical characterization in the Kuramoto case. Our results propose a self-organized approach to design and control abrupt transitions in adaptive biological systems and electronic circuits, and place explosive synchronization and percolation under the same mechanistic framework.