  A conjecture of Erd\H{o}s from 1967 asserts that any graph on $n$ vertices
which does not contain a fixed $r$-degenerate bipartite graph $F$ has at most
$Cn^{2-1/r}$ edges, where $C$ is a constant depending only on $F$. We show that
this bound holds for a large family of $r$-degenerate bipartite graphs,
including all $r$-degenerate blow-ups of trees. Our results generalise many
previously proven cases of the Erd\H{o}s conjecture, including the related
results of F\"uredi and Alon, Krivelevich and Sudakov. Our proof uses
supersaturation and a random walk on an auxiliary graph.
