A long-standing but fundamental question in lightning studies concerns how lightning is initiated inside storms, given the absence of physical conductors. The issue has revolved around the question of whether the discharges are initiated solely by conventional dielectric breakdown or involve relativistic runaway electron processes. Here we report observations of a relatively unknown type of discharge, called fast positive breakdown, that is the cause of high-power discharges known as narrow bipolar events. The breakdown is found to have a wide range of strengths and is the initiating event of numerous lightning discharges. It appears to be purely dielectric in nature and to consist of a system of positive streamers in a locally intense electric field region. It initiates negative breakdown at the starting location of the streamers, which leads to the ensuing flash. The observations show that many or possibly all lightning flashes are initiated by fast positive breakdown.