Weakly bounded height on modular curves

We study the intersection of a fixed plane algebraic curve C with modular curves of varying level. The height of points in such intersections cannot be bounded from above independently of the level when C is defined over the field of algebraic numbers. But we find a certain class of curves C for which the height is bounded logarithmically in the level. This bound is strong enough to imply certain finiteness result. Such evidence leads to a conjecture involving a logarithmic height bound unless C is of so-called special type. We also discuss connections to recent progress on conjectures concerning unlikely intersections.