We investigate how the complexity of Euclidean TSP for point sets P inside the strip (−∞,+∞)×[0,δ] depends on the strip width δ. We obtain two main results. First, for the case where the points have distinct integer x-coordinates, we prove that a shortest bitonic tour (which can be computed in O(nlog2n) time using an existing algorithm) is guaranteed to be a shortest tour overall when δ≤22–√, a bound which is best possible. Second, we present an algorithm that is fixed-parameter tractable with respect to δ. More precisely, our algorithm has running time 2O(δ√)n2 for sparse point sets, where each 1×δ rectangle inside the strip contains O(1) points. For random point sets, where the points are chosen uniformly at random from the rectangle~[0,n]×[0,δ], it has an expected running time of 2O(δ√)n2+O(n3).