It has been proposed that the observed systems of hot super-Earths formed in situ from high-mass discs. By fitting a disc profile to the entire population of Kepler planet candidates, Chiang & Laughlin constructed a 'minimum-mass extrasolar nebula' with surface density profile Σ ∝ r −1.6. Here, we use multiple-planet systems to show that it is inconsistent to assume a universal disc profile. Systems with 3-6 low-mass planets (or planet candidates) produce a diversity of minimum-mass discs with surface density profiles ranging from Σ ∝ r −3.2 to Σ ∝ r 0.5 (5th-95th percentile). By simulating the transit detection of populations of synthetic planetary systems designed to match the properties of observed super-Earth systems, we show that a universal disc profile is statistically excluded at high confidence. Rather, the underlying distribution of minimum-mass discs is characterized by a broad range of surface density slopes. Models of gaseous discs can only explain a narrow range of slopes (roughly between r 0 and r −1.5). Yet accretion of terrestrial planets in a gas-free environment preserves the initial radial distribution of building blocks. The known systems of hot super-Earths must therefore not represent the structure of their parent gas discs and cannot have predominantly formed in situ. We instead interpret the diversity of disc slopes as the imprint of a process that re-arranged the solids relative to the gas in the inner parts of protoplanetary discs. A plausible mechanism is inward type 1 migration of Mars- to Earth-mass planetary embryos, perhaps followed by a final assembly phase.