  One of the sources of incompatibility between general relativity and quantum
mechanics is perturbative non-renormalizability of quantum gravity in $3+1$
spacetime dimensions. Here, we show that in the presence of disorder induced by
random networks of observers measuring covariant quantities (such as scalar
curvature) $(3+1)$-dimensional quantum gravity exhibits an effective
dimensional reduction at large spatio-temporal scales, which is analogous to
the Parisi-Sourlas phenomenon observed for quantum field theories in random
external fields. After averaging over associated disorder and focusing on the
infrared dynamics of the theory we find that the upper critical dimension of
quantum gravity is lifted from $D_{{\rm cr}}=1+1$ to $D_{{\rm cr}}=3+1$
dimensions.
