  It is known that for subgraph-closed graph classes the first-order model
checking problem is fixed-parameter tractable if and only if the class is
nowhere dense [Grohe, Kreutzer, Siebertz, STOC 2014]. However, the dependency
on the formula size is non-elementary, and in fact, this is unavoidable even
for the class of all trees [Frick and Grohe, LICS 2002]. On the other hand, it
is known that the dependency is elementary for classes of bounded degree [Frick
and Grohe, LICS 2002] as well as for classes of bounded pathwidth [Lampis,
ICALP 2023]. In this paper we generalise these results and almost completely
characterise subgraph-closed graph classes for which the model checking problem
is fixed-parameter tractable with an elementary dependency on the formula size.
Those are the graph classes for which there exists a number $d$ such that for
every $r$, some tree of depth $d$ and size bounded by an elementary function of
$r$ is avoided as an $({\leq} r)$-subdivision in all graphs in the class. In
particular, this implies that if the class in question excludes a fixed tree as
a topological minor, then first-order model checking for graphs in the class is
fixed-parameter tractable with an elementary dependency on the formula size.
