  A graph $G$ contains a graph $H$ as an induced minor if $H$ can be obtained
from $G$ after vertex deletions and edge contractions. We show that for every
$k$-vertex planar graph $H$, every graph $G$ excluding $H$ as an induced minor
and $K_{t,t}$ as a subgraph has treewidth at most $\Delta(G)^{f(k,t)}$ where
$\Delta(G)$ denotes the maximum degree of $G$. Without requiring the absence of
a $K_{t,t}$ subgraph, Korhonen [JCTB '23] has shown the upper bound of
$k^{O(1)} 2^{\Delta(G)^5}$ whose dependence in $\Delta(G)$ is exponential. Our
result partially answers a question of Chudnovsky [Dagstuhl seminar '23] asking
whether the treewidth of graphs with $\Delta(G)=O(\log{|V(G)|})$ excluding both
a $k$-vertex planar graph as an induced minor and the biclique $K_{t,t}$ as a
subgraph is in $O_{k,t}(\log |V(G)|)$. We confirm that the treewidth is in this
case polylogarithmic in $|V(G)|$.
