  In this paper we show that every graph of pathwidth less than $k$ that has a
path of order $n$ also has an induced path of order at least $\frac{1}{3}
n^{1/k}$. This is an exponential improvement and a generalization of the
polylogarithmic bounds obtained by Esperet, Lemoine and Maffray (2016) for
interval graphs of bounded clique number. We complement this result with an
upper-bound.
  This result is then used to prove the two following generalizations:
  - every graph of treewidth less than $k$ that has a path of order $n$
contains an induced path of order at least $\frac{1}{4} (\log n)^{1/k}$;
  - for every non-trivial graph class that is closed under topological minors
there is a constant $d \in (0,1)$ such that every graph from this class that
has a path of order $n$ contains an induced path of order at least $(\log
n)^d$.
  We also describe consequences of these results beyond graph classes that are
closed under topological minors.
