  The popular (piecewise) quadratic schemes for the biharmonic equation based
on triangles are the nonconforming Morley finite element, the discontinuous
Galerkin, the $C^0$ interior penalty, and the WOPSIP schemes. Those methods are
modified in their right-hand side $F\in H^{-2}(\Omega)$ replaced by $F\circ
(JI_{\rm M}) $ and then are quasi-optimal in their respective discrete norms.
The smoother $JI_{\rm M}$ is defined for a piecewise smooth input function by a
(generalized) Morley interpolation $I_{\rm M}$ followed by a companion operator
$J$. An abstract framework for the error analysis in the energy, weaker and
piecewise Sobolev norms for the schemes is outlined and applied to the
biharmonic equation. Three errors are also equivalent in some particular
discrete norm from [Carstensen, Gallistl, Nataraj: Comparison results of
nonstandard $P_2$ finite element methods for the biharmonic problem, ESAIM
Math. Model. Numer. Anal. (2015)] without data oscillations. This paper extends
the work [Veeser, Zanotti: Quasi-optimal nonconforming methods for symmetric
elliptic problems, SIAM J. Numer. Anal. 56 (2018)] to the discontinuous
Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.
