  We devise new variants of the following nonconforming finite element methods:
DG methods of fixed arbitrary order for the Poisson problem, the
Crouzeix-Raviart interior penalty method for linear elasticity, and the
quadratic $C^0$ interior penalty method for the biharmonic problem. Each
variant differs from the original method only in the discretization of the
right-hand side. Before applying the load functional, a linear operator
transforms nonconforming discrete test functions into conforming functions such
that stability and consistency are improved. The new variants are thus
quasi-optimal with respect to an extension of the energy norm. Furthermore,
their quasi-optimality constants are uniformly bounded for shape regular meshes
and tend to $1$ as the penalty parameter increases.
