  We point out that the Cartan geometry known as the second-order conformal
structure provides a natural differential geometric framework underlying gauge
theories of conformal gravity. We are concerned by two theories: the first one
will be the associated Yang-Mills-like Lagrangian, while the second, inspired
by~\cite{Wheeler2014}, will be a slightly more general one which will relax the
conformal Cartan geometry. The corresponding gauge symmetry is treated within
the BRST language. We show that the Weyl gauge potential is a spurious degree
of freedom, analogous to a Stueckelberg field, that can be eliminated through
the dressing field method. We derive sets of field equations for both the
studied Lagrangians. For the second one, they constrain the gauge field to be
the `normal conformal Cartan connection'. Finally, we provide in a Lagrangian
framework a justification of the identification, in dimension $4$, of the Bach
tensor with the Yang-Mills current of the normal conformal Cartan connection,
as proved in \cite{Korz-Lewand-2003}.
