  For any small positive real $\varepsilon$ and integer $t >
\frac{1}{\varepsilon}$, we build a graph with a vertex deletion set of size $t$
to a tree, and twin-width greater than $2^{(1-\varepsilon) t}$. In particular,
this shows that the twin-width is sometimes exponential in the treewidth, in
the so-called oriented twin-width and grid number, and that adding an apex may
multiply the twin-width by at least $2-\varepsilon$. Except for the one in
oriented twin-width, these lower bounds are essentially tight.
