  We consider one-sided error property testing of $\mathcal{F}$-minor freeness
in bounded-degree graphs for any finite family of graphs $\mathcal{F}$ that
contains a minor of $K_{2,k}$, the $k$-circus graph, or the $(k\times 2)$-grid
for any $k\in\mathbb{N}$. This includes, for instance, testing whether a graph
is outerplanar or a cactus graph. The query complexity of our algorithm in
terms of the number of vertices in the graph, $n$, is $\tilde{O}(n^{2/3} /
\epsilon^5)$. Czumaj et~al.\ showed that cycle-freeness and $C_k$-minor
freeness can be tested with query complexity $\tilde{O}(\sqrt{n})$ by using
random walks, and that testing $H$-minor freeness for any $H$ that contains a
cycles requires $\Omega(\sqrt{n})$ queries. In contrast to these results, we
analyze the structure of the graph and show that either we can find a subgraph
of sublinear size that includes the forbidden minor $H$, or we can find a pair
of disjoint subsets of vertices whose edge-cut is large, which induces an
$H$-minor.
