The spreading of contagions can exhibit a percolation transition, which separates transitory prevalence from outbreaks that reach a finite fraction of the population1,2. Such transitions are commonly believed to be continuous, but empirical studies have shown more violent spreading modes when the participating agents are not limited to one type. Striking examples include the co-epidemic of the Spanish flu and pneumonia that occurred in 1918 (refs 3, 4), and, more recently, the concurrent prevalence of HIV/AIDS and a host of diseases5,6,7. It remains unclear to what extent an outbreak in the presence of interacting pathogens differs from that due to an ordinary single-agent process. Here we study a mechanistic model for understanding contagion processes involving inter-agent cooperation. Our stochastic simulations reveal the possible emergence of a massive avalanche-like outbreak right at the threshold, which is manifested as a discontinuous phase transition. Such an abrupt change arises only if the underlying network topology supports a bottleneck for cascaded mutual infections. Surprisingly, all these discontinuous transitions are accompanied by non-trivial critical behaviours, presenting a rare case of hybrid transition8. The findings may imply the origin of catastrophic occurrences in many realistic systems, from co-epidemics to financial contagions9.