  We construct classes of graphs that are variants of the so-called layered
wheel. One of their key properties is that while the treewidth is bounded by a
function of the clique number, the construction can be adjusted to make the
dependance grow arbitrarily. Some of these classes provide counter-examples to
several conjectures. In particular, the construction includes hereditary
classes of graphs whose treewidth is bounded by a function of the clique number
while the tree-independence number is unbounded, thus disproving a conjecture
of Dallard, Milani\v{c} and \v{S}torgel [Treewidth versus clique number. II.
Tree-independence number. Journal of Combinatorial Theory, Series B,
164:404-442, 2024.]. The construction can be further adjusted to provide, for
any fixed integer $c$, graphs of arbitrarily large treewidth that contain no
$K_c$-free graphs of high treewidth, thus disproving a conjecture of Hajebi
[Chordal graphs, even-hole-free graphs and sparse obstructions to bounded
treewidth, arXiv:2401.01299, 2024].
