  We consider a natural generalisation of Tur\'an's forbidden subgraph problem
and the Ruzsa-Szemer\'edi problem by studying the maximum number $ex_F(n,G)$ of
edge-disjoint copies of a fixed graph $F$ can be placed on an $n$-vertex ground
set without forming a subgraph $G$ whose edges are from different $F$-copies.
We determine the pairs $\{F, G\}$ for which the order of magnitude of
$ex_F(n,G)$ is quadratic and prove several asymptotic results using various
tools from the regularity lemma and supersaturation to graph packing results.
