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We revisit the picture of jets propagating in the quark-gluon plasma. In addition to vacuum radiation, partons scatter on the medium constituents resulting in induced emissions. Analytical approaches to including these interactions have traditionally dealt separately with multiple, soft, or rare, hard scatterings. A full description has so far only been available using numerical methods. We achieve full analytical control of the relevant scales and map out the dominant physical processes in the full phase space. To this aim, we extend existing expansion schemes for the medium-induced spectrum to the Bethe--Heitler regime. This covers the whole phase space from early to late times, and from hard splittings to emissions below the thermal scale. Based on the separation of scales, a space-time picture naturally emerges: at early times, induced emissions start to build from rare scatterings with the medium. At a later stage, induced emissions due to multiple soft scatterings result in a turbulent cascade that rapidly degrades energy down to, and including, the Bethe--Heitler regime. We quantify the impact of such an improved picture, compared to the current state-of-the-art factorization that includes only soft scatterings, by both analytical and numerical methods for the medium-induced energy distribution function. Our work serves to improve our understanding of jet quenching from small to large systems and for future upgrades of Monte Carlo generators.