  We consider sequences of symmetric $U$-statistics, not necessarily
Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove
quantitative central limit theorems (CLTs) based on the use of {\it contraction
operators}. Our results represent an explicit counterpart to analogous criteria
that are available for sequences of random variables living on the Gaussian,
Poisson or Rademacher chaoses, and are perfectly tailored for geometric
applications. As a demonstration of this fact, we develop explicit bounds for
subgraph counting in generalised random graphs on Euclidean spaces; special
attention is devoted to the so-called `dense parameter regime' for uniformly
distributed points, for which we deduce CLTs that are new even in their
qualitative statement, and that substantially extend classical findings by
Jammalamadaka and Janson (1986) and Bhattacharaya and Ghosh (1992).
