We provide theoretical insights around the cutwidth of a graph and the One-Sided Crossing Minimization (OSCM) problem. OSCM was posed in the Parameterized Algorithms and Computational Experiments Challenge 2024, where the cutwidth of the input graph was the parameter in the parameterized track. We prove an asymptotically sharp upper bound on the size of a graph in terms of its order and cutwidth. As the number of so-called unsuited pairs is one of the factors that determine the difficulty of an OSCM instance, we provide a sharp upper bound on them in terms of the order $n$ and the cutwidth of the input graph. If the cutwidth is bounded by a constant, this implies an $\mathcal{O}(2^n)$-time algorithm, while the trivial algorithm has a running time of $\mathcal{O}(2^{n^2})$. At last, we prove structural properties of the so-called crossing numbers in an OSCM instance.