  Weak galaxy lensing surveys have consistently reported a lower amplitude for
the matter fluctuation spectrum, as measured by the $S_8$ parameter, than
expected in the $\Lambda{\rm CDM}$ cosmology favoured by $Planck$. However, the
expansion history follows the predictions of the $Planck$ $\Lambda{\rm CDM}$
cosmology to high accuracy, as do measurements of lensing of the cosmic
microwave background anisotropies. Redshift space distortion measurements also
appear to be consistent with $Planck$ $\Lambda{\rm CDM}$. In this paper, we
argue that these observations can be reconciled with the $Planck$ $\Lambda{\rm
CDM}$ cosmology if the matter power spectrum is suppressed more strongly on
non-linear scales than assumed in analyses of weak galaxy lensing. We
demonstrate this point by fitting a one-parameter model, characterising a
suppression of the non-linear power spectrum, to the KiDS-1000 weak lensing
measurements. Such a suppression could be attributed to new properties of the
dark matter that affect non-linear scales, or to a response of the matter
fluctuations to baryonic feedback processes that are stronger than expected
from recent cosmological simulations. Our proposed explanation can be tested
using measurements of the amplitude of the matter fluctuation spectrum on
linear scales, in particular via high precision redshift space distortion
measurements from forthcoming galaxy and quasar redshift surveys.
