We present a comparative analysis of the CH + (1-0) and Lyα lines, observed with the Atacama Large Millimeter Array and Keck telescope, respectively, in the field of the submillimetre-selected galaxy SMM J02399−0136 at z ∼ 2.8, which comprises a heavily obscured starburst galaxy and a broad absorption line quasar, immersed in a large Lyα nebula. This comparison highlights the critical role played by turbulence in channelling the energy across gas phases and scales, splitting the energy trail between hot/thermal and cool/turbulent phases in the circumgalactic medium (CGM). The unique chemical and spectroscopic properties of CH + are used to infer the existence of a massive (∼3.5 × 10 10 M), highly turbulent reservoir of diffuse molecular gas of radius ∼20 kpc coinciding with the core of the Lyα nebula. The whole cool and cold CGM is shown to be inflowing towards the galaxies at a velocity ∼ 400 km s −1. Several kpc-scale shocks are detected tentatively in CH + emission. Their linewidth and specific location in space and velocity with respect to the high-velocity Lyα emission suggest that they lie at the interface of the inflowing CGM and the high-velocity outflowing gas. They signpost the feeding of CGM turbulence by active galactic nucleiand stellar-driven outflows. The mass and energy budgets of the CGM require net mass accretion at a rate commensurate with the star formation rate. From this similarity, we infer that the merger-driven burst of star formation and black-hole growth are ultimately fuelled by large-scale gas accretion.