  An extension $B\subset A$ of finite dimensional algebras is bounded if the
$B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is
finite and $\mathrm{Tor}_i^B(A/B, (A/B)^{\otimes_B j})=0$ for all $i, j\geq 1$.
We show that for a bounded extension $B\subset A$, the algebras $A$ and $B$ are
singularly equivalent of Morita type with level. Additionally, under mild
conditions, their stable categories of Gorenstein projective modules and
Gorenstein defect categories are equivalent, respectively. Some homological
conjectures are also investigated for bounded extensions, including
Auslander-Reiten conjecture, finististic dimension conjecture, Fg condition,
Han's conjecture, and Keller's conjecture. Applications to trivial extensions
and triangular matrix algebras are given.
  In course of proof, we give some handy criteria for a functor between module
categories to induce triangle functors between stable categories of Gorenstein
projective modules and Gorenstein defect categories, which generalise some
known criteria, and hence might be of independent interest.
