During magnetic reconnection, the field lines must break and reconnect to release the energy that drives solar and stellar flares(1,2) and other explosive events in space(3) and in the laboratory(4). Exactly how this happens has been unclear, because dissipation is needed to break magnetic field lines and classical collisions are typically weak. Ion-electron drag arising from turbulence(5), dubbed 'anomalous resistivity', and thermal momentum transport(6) are two mechanisms that have been widely invoked. Measurements of enhanced turbulence near reconnection sites in space(7,8) and in the laboratory(9,10) support the anomalous resistivity idea but there has been no demonstration from measurements that this turbulence produces the necessary enhanced drag(11). Here we report computer simulations that show that neither of the two previously favoured mechanisms controls how magnetic field lines reconnect in the plasmas of greatest interest, those in which the magnetic field dominates the energy budget. Rather, we find that when the current layers that form during magnetic reconnection become too intense, they disintegrate and spread into a complex web of filaments that causes the rate of reconnection to increase abruptly. This filamentary web can be explored in the laboratory or in space with satellites that can measure the resulting electromagnetic turbulence.