We give essentially tight bounds for, ν(d, k), the maximum number of distinct neighbourhoods on a set X of k vertices in a graph with twin-width at most d. Using the celebrated Marcus-Tardos theorem, two independent works [Bonnet et al., Algorithmica '22; Przybyszewski '22] have shown the upper bound ν(d, k) ⩽ exp(exp(O(d)))k, with a double-exponential dependence in the twin-width. The work of [Gajarsky et al., ICALP '22], using the framework of local types, implies the existence of a single-exponential bound (without explicitly stating such a bound). We give such an explicit bound, and prove that it is essentially tight. Indeed, we give a short self-contained proof that for every d and k ν(d, k) ⩽ (d + 2)2 d+1 k = 2 d+log d+Θ(1) k, and build a bipartite graph implying ν(d, k) ⩾ 2 d+log d+Θ(1) k, in the regime when k is large enough compared to d.