‘Long’ γ-ray bursts (GRBs) are commonly accepted to originate in the explosion of particularly massive stars, which give rise to highly relativistic jets. Inhomogeneities in the expanding flow result in internal shock waves that are believed to produce the γ-rays we see^,. As the jet travels further outward into the surrounding circumstellar medium, ‘external’ shocks create the afterglow emission seen in the X-ray, optical and radio bands^,. Here we report observations of the early phases of the X-ray emission of five GRBs. Their X-ray light curves are characterised by a surprisingly rapid fall-off for the first few hundred seconds, followed by a less rapid decline lasting several hours. This steep decline, together with detailed spectral properties of two particular bursts, shows that violent shock interactions take place in the early jet outflows.