ABSTRACT The Ly α forest is the large-scale structure probe for which we appear to have modelling control to the highest wavenumbers. This makes the Ly α forest of great interest for constraining the warmness/fuzziness of dark matter and the timing of reionization processes. However, the standard statistic, the Ly α forest power spectrum, is unable to strongly constrain the intergalactic medium (IGM) temperature–density relation, and this inability further limits how well other high-wavenumber-sensitive parameters can be constrained. With the aim of breaking these degeneracies, we measure the power spectrum of the Ly β forest and its cross-correlation with the coeval Ly α forest using the 100 spectra of z = 3.5–4.5 quasars in the VLT/X-Shooter XQ-100 Legacy Survey, motivated by the Ly β transition’s smaller absorption cross-section that makes it sensitive to somewhat higher densities relative to the Ly α transition. Our inferences from this measurement for the IGM temperature–density relation appear to latch consistently on to the recent tight lower redshift Ly α forest constraints. The z = 3.4–4.7 trends we find using the Ly α–Ly β cross-correlation show a flattening of the slope of the temperature–density relation with decreasing redshift. This is the trend anticipated from ongoing He ii reionization and there being sufficient time to reach the asymptotic temperature–density slope after hydrogen reionization completes. Furthermore, our measurements provide a consistency check on IGM models that explain the Ly α forest, with the cross-correlation being immune to systematics that are uncorrelated between the two forests, such as metal line contamination.