Twin-width can be exponential in treewidth

For any small positive real ε and integer t>1ε, we build a graph with a vertex deletion set of size t to a tree, and twin-width greater than 2(1−ε)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−ε. Except for the one in oriented twin-width, these lower bounds are essentially tight.