The systematic study of Turán-type extremal problems for edge-ordered graphs was initiated by Gerbner et al. arXiv:2001.00849. They conjectured that the extremal functions of edge-ordered forests of order chromatic number 2 are $n^{1+o(1)}$. Here we resolve this conjecture proving the stronger upper bound of $n2^{O(\sqrt{\log n})}$. This represents a gap in the family of possible extremal functions as other forbidden edge-ordered graphs have extremal functions $Ω(n^c)$ for some $c>1$. However, our result is probably not the last word: here we conjecture that the even stronger upper bound of $n\log^{O(1)}n$ also holds for the same set of extremal functions.