Let \(C\) be a smooth curve of genus \(g\) with gonality \(d\). The gonality conjecture (or Green-Lazarsfeld conjecture) predicts the minimal free resolution of large degree linearly normal embeddings of \(C\) in projective spaces see [M. Aprodu and J. Nagel, Koszul cohomology and algebraic geometry. University Lecture Series 52. Providence, RI: American Mathematical Society (AMS). (2010; Zbl 1189.14001)]. It is known in several cases, e.g. for general \(d\)-gonal curves M. Aprodu [Math. Res. Lett. 12, 387–400 (2005; Zbl 1084.14032)], curves in Hirzebruch surfaces M. Aprodu [Math. Z. 241, 1–15 (2002; Zbl 1037.14012)] and on some toric surfaces R. Kawaguchi [Osaka J. Math. 45, No. 1, 113–126 (2008; Zbl 1133.14306)]. Here the author proves it for curves in toric surfaces with two \(\mathbb {P}^1\)-fibrations (under some conditions).