For a graph
$G$, let
$f(G)$ denote the maximum number of edges in a bipartite subgraph of
$G$. Given a fixed graph
$H$ and a positive integer
$m$, let
$f(m,H)$ denote the minimum possible cardinality of
$f(G)$, as
$G$ ranges over all graphs on
$m$ edges that contain no copy of
$H$. Alon et al. [‘Maximum cuts and judicious partitions in graphs without short cycles’, J. Combin. Theory Ser. B 88 (2003), 329–346] conjectured that, for any fixed graph
$H$, there exists an
$\unicode[STIX]{x1D716}(H)>0$ such that
$f(m,H)\geq m/2+\unicode[STIX]{x1D6FA}(m^{3/4+\unicode[STIX]{x1D716}})$. We show that, for any wheel graph
$W_{2k}$ of
$2k$ spokes, there exists
$c(k)>0$ such that
$f(m,W_{2k})\geq m/2+c(k)m^{(2k-1)/(3k-1)}\log m$. In particular, we confirm the conjecture asymptotically for
$W_{4}$ and give general lower bounds for
$W_{2k+1}$.