  We investigate the twin-width of the Erd\H{o}s-R\'enyi random graph $G(n,p)$.
We unveil a surprising behavior of this parameter by showing the existence of a
constant $p^*\approx 0.4$ such that with high probability, when $p^*\le p\le
1-p^*$, the twin-width is asymptotically $2p(1-p)n$, whereas, when $0<p<p^*$ or
$1>p>1-p^*$, the twin-width is significantly higher than $2p(1-p)n$. In
addition, we show that the twin-width of $G(n,1/2)$ is concentrated around $n/2
- \sqrt{3n \log n}/2$ within an interval of length $o(\sqrt{n\log n})$. For the
sparse random graph, we show that with high probability, the twin-width of
$G(n,p)$ is $\Theta(n\sqrt{p})$ when $(726\ln n)/n\leq p\leq1/2$.
