We give essentially tight bounds for, $\nu(d,k)$, 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 $\nu(d,k) \leqslant \exp(\exp(O(d)))k$, with a double-exponential dependence in the twin-width. We give a short self-contained proof that for every $d$ and $k$, $$\nu(d,k) \leqslant (d+2)2^{d+1}k = 2^{d+O(\log d)}k,$$ and build a bipartite graph implying $\nu(d,k) \geqslant 2^{d+\log d+O(1)}k$, in the regime when $k$ is large enough compared to $d$.