Bright quasars, powered by accretion onto billion-solar-mass black holes, already existed at the epoch of reionization, when the Universe was 0.5-1 billion years old. How these black holes formed in such a short time is the subject of debate, particularly as they lie above the correlation between black-hole mass and galaxy dynamical mass in the local Universe. What slowed down black-hole growth, leading towards the symbiotic growth observed in the local Universe, and when this process started, has hitherto not been known, although black-hole feedback is a likely driver. Here we report optical and near-infrared observations of a sample of quasars at redshifts 5.8 ≲ z ≲ 6.6. About half of the quasar spectra reveal broad, blueshifted absorption line troughs, tracing black-hole-driven winds with extreme outflow velocities, up to 17% of the speed of light. The fraction of quasars with such outflow winds at z ≳ 5.8 is ≈2.4 times higher than at z ≈ 2-4. We infer that outflows at z ≳ 5.8 inject large amounts of energy into the interstellar medium and suppress nuclear gas accretion, slowing down black-hole growth. The outflow phase may then mark the beginning of substantial black-hole feedback. The red optical colours of outflow quasars at z ≳ 5.8 indeed suggest that these systems are dusty and may be caught during an initial quenching phase of obscured accretion.