  We devise and analyze $C^0$-conforming hybrid high-order (HHO) methods to
approximate biharmonic problems with either clamped or simply supported
boundary conditions. $C^0$-conforming HHO methods hinge on cell unknowns which
are $C^0$-conforming polynomials of order $(k+2)$ approximating the solution in
the mesh cells and on face unknowns which are polynomials of order $k\ge0$
approximating the normal derivative of the solution on the mesh skeleton. Such
methods deliver $O(h^{k+1})$ $H^2$-error estimates for smooth solutions. An
important novelty in the error analysis is to lower the minimal regularity
requirement on the exact solution. The technique to achieve this has broader
applicability than just $C^0$-conforming HHO methods, and to illustrate this
point, we outline the error analysis for the well-known $C^0$-conforming
interior penalty discontinuous Galerkin (IPDG) methods as well. The present
technique does not require bubble functions or a $C^1$-smoother to evaluate the
right-hand side in case of rough loads. Finally, numerical results including
comparisons to various existing methods showcase the efficiency of the proposed
$C^0$-conforming HHO methods.
