  The systematic study of Tur\'an-type extremal problems for edge-ordered
graphs was initiated by Gerbner et al. in 2020. Here we characterize connected
edge-ordered graphs with linear extremal functions and show that the extremal
function of other connected edge-ordered graphs is $\Omega(n\log n)$. This
characterization and dichotomy are similar in spirit to results of F\"uredi et
al. (2020) about vertex-ordered and convex geometric graphs. We also extend the
study of extremal function of short edge-ordered paths by Gerbner et al. to
some longer paths.
