  We continue the study of $(\mathrm{tw},\omega)$-bounded graph classes, that
is, hereditary graph classes in which the treewidth can only be large due to
the presence of a large clique, with the goal of understanding the extent to
which this property has useful algorithmic implications for the Independent Set
and related problems. In the previous paper of the series [Dallard,
Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. II.
Tree-independence number], we introduced the tree-independence number, a
min-max graph invariant related to tree decompositions. Bounded
tree-independence number implies both $(\mathrm{tw},\omega)$-boundedness and
the existence of a polynomial-time algorithm for the Maximum Weight Independent
Set problem, provided that the input graph is given together with a tree
decomposition with bounded independence number.
  In this paper, we consider six graph containment relations and for each of
them characterize the graphs $H$ for which any graph excluding $H$ with respect
to the relation admits a tree decomposition with bounded independence number.
The induced minor relation is of particular interest: we show that excluding
either a $K_5$ minus an edge or the $4$-wheel implies the existence of a tree
decomposition in which every bag is a clique plus at most $3$ vertices, while
excluding a complete bipartite graph $K_{2,q}$ implies the existence of a tree
decomposition with independence number at most $2(q-1)$. Our constructive
proofs are obtained using a variety of tools, including $\ell$-refined tree
decompositions, SPQR trees, and potential maximal cliques. They imply
polynomial-time algorithms for the Independent Set and related problems in an
infinite family of graph classes; in particular, the results apply to the class
of $1$-perfectly orientable graphs, answering a question of Beisegel,
Chudnovsky, Gurvich, Milani\v{c}, and Servatius from 2019.
