  We show that the labeled Thompson groups and the twisted Brin--Thompson
groups are boundedly acyclic. This allows us to prove several new embedding
results for groups. First, every group of type $F_n$ embeds quasi-isometrically
into a boundedly acyclic group of type $F_n$ that has no proper finite index
subgroups. This improves a result of Bridson \cite{Br98} and a theorem of
Fournier-Facio--L\"oh--Moraschini \cite[Theorem 2]{FFCM21}. Second, every group
of type $F_n$ embeds quasi-isometrically into a $5$-uniformly perfect group of
type $F_n$. Third, using Belk--Zaremsky's construction of twisted
Brin--Thompson groups, we show that every finitely generated group embeds
quasi-isometrically into a finitely generated boundedly acyclic simple group.
