  The \emph{Tur\'an function} $\ex(n,F)$ of a graph $F$ is the maximum number
of edges in an $F$-free graph with $n$ vertices. The classical results of
Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs
where the key question is to determine $h_F(n,q)$, the minimum number of copies
of $F$ that a graph with $n$ vertices and $\ex(n,F)+q$ edges can have.
  We determine $h_F(n,q)$ asymptotically when $F$ is \emph{color-critical}
(that is, $F$ contains an edge whose deletion reduces its chromatic number) and
$q=o(n^2)$.
  Determining the exact value of $h_F(n,q)$ seems rather difficult. For
example, let $c_1$ be the limit superior of $q/n$ for which the extremal
structures are obtained by adding some $q$ edges to a maximum $F$-free graph.
The problem of determining $c_1$ for cliques was a well-known question of Erd\H
os that was solved only decades later by Lov\'asz and Simonovits. Here we prove
that $c_1>0$ for every {color-critical}~$F$. Our approach also allows us to
determine $c_1$ for a number of graphs, including odd cycles, cliques with one
edge removed, and complete bipartite graphs plus an edge.
