  In convergence analysis of finite element methods for singularly perturbed
reaction--diffusion problems, balanced norms have been successfully introduced
to replace standard energy norms so that layers can be captured. In this
article, we focus on the convergence analysis in a balanced norm on
Bakhvalov-type rectangular meshes. In order to achieve our goal, a novel
interpolation operator, which consists of a local weighted $L^2$ projection
operator and the Lagrange interpolation operator, is introduced for a
convergence analysis of optimal order in the balanced norm. The analysis also
depends on the stabilities of the $L^2$ projection and the characteristics of
Bakhvalov-type meshes. Furthermore, we obtain a supercloseness result in the
balanced norm, which appears in the literature for the first time. This result
depends on another novel interpolant, which consists of the local weighted
$L^2$ projection operator, a vertices-edges-element operator and some
corrections on the boundary.
