An ordered graph H is a simple graph with a linear order on its vertex set. The corresponding Turan problem, first studied by Pach and Tardos, asks for the maximum number 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 ex(<) (n, H) > n(1+epsilon) for some positive epsilon = epsilon(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 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. (C) 2019 Elsevier Inc. All rights reserved.