  An ordered graph $H$ is a simple graph with a linear order on its vertex set.
The corresponding Tur\'an problem, first studied by Pach and Tardos, asks for
the maximum number $\text{ex}_<(n,H)$ of edges in an ordered graph on $n$
vertices that does not contain $H$ as an ordered subgraph. It is known that
$\text{ex}_<(n,H) > n^{1+\varepsilon}$ for some positive
$\varepsilon=\varepsilon(H)$ unless $H$ is a forest that has a proper
2-coloring with one color class totally preceding the other one. Making
progress towards a conjecture of Pach and Tardos, we prove that
$\text{ex}_<(n,H) =n^{1+o(1)}$ holds for all such forests that are "degenerate"
in a certain sense. This class includes every forest for which an $n^{1+o(1)}$
upper bound was previously known, as well as new examples. Our proof is based
on a density-increment argument.
