ABSTRACT We present the results from an X-ray variability study of IRAS 13224-3809. This is probably the best source for X-ray reverberation studies since it is X-ray bright, extremely variable, and it has been extensively observed with XMM–Newton. We used all the archival XMM–Newton data from the three EPIC cameras (to increase the signal to noise) and, given the many observations of the source, we were able to compute the time lags spectra in three different flux levels/periods. We fitted the time lags and energy spectra, simultaneously, using a new X-ray reverberation code that computes the time-dependent reflection spectra of the disc as a response to an X-ray flash from a point source located on the axis of the black hole (BH) accretion disc (lamp-post geometry). To the best of our knowledge, this is the first time for active galactic nuclei that both time lags and energy spectra are fitted by a model simultaneously in different flux periods. The model fits in the case when the BH is rapidly rotating are significantly better than the model fits in the case of a Schwarzschild BH. This result strongly favours the hypothesis of a rotating central BH in this source. We also detect significant variations in the height of the X-ray corona. The X-ray height appears to increase from ∼3–5 gravitational radii when the X-ray luminosity is of the order of ∼1.5–3 per cent of the Eddington limit, up to ∼10 gravitational radii, when the luminosity doubles.