  We prove that relational structures admitting specific polymorphisms (namely,
canonical pseudo-WNU operations of all arities $n \geq 3$) have low relational
width. This implies a collapse of the bounded width hierarchy for numerous
classes of infinite-domain CSPs studied in the literature. Moreover, we obtain
a characterization of bounded width for first-order reducts of unary structures
and a characterization of MMSNP sentences that are equivalent to a Datalog
program, answering a question posed by Bienvenu, ten Cate, Lutz, and Wolter. In
particular, the bounded width hierarchy collapses in those cases as well. Our
results extend the scope of theorems of Barto and Kozik characterizing bounded
width for finite structures, and show the applicability of infinite-domain CSPs
to other fields.
