Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most k. We prove that every minimal immersion obstruction for cutwidth at most k has size at most 2 O(k 3 log k). As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time 2 O(k 2 log k) · n, where k is the optimum width and n is the number of vertices. While being slower by a log k-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1-24, 2005] and [Cutwidth II: Algorithms for partial w-trees of bounded degree, J. Algorithms, 56(1):25-49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.