  In even spacetime dimensions, the interacting bosonic conformal higher-spin
(CHS) theory can be realised as an induced action. The main ingredient in this
definition is the model $\mathcal{S}[\varphi,h]$ describing a complex scalar
field $\varphi$ coupled to an infinite set of background CHS fields $h$, with
$\mathcal{S}[\varphi,h]$ possessing a non-abelian gauge symmetry. Two
characteristic features of the perturbative constructions of
$\mathcal{S}[\varphi , h]$ given in the literature are: (i) the background
spacetime is flat; and (ii) conformal invariance is not manifest. In the
present paper we provide a new derivation of this action in four dimensions
such that (i) $\mathcal{S}[\varphi , h]$ is defined on an arbitrary
conformally-flat background; and (ii) the background conformal symmetry is
manifestly realised. Next, our results are extended to the $\mathcal{N}=1$
supersymmetric case. Specifically, we construct, for the first time, a model
$\mathcal{S}[\Phi, H]$ for a conformal scalar/chiral multiplet $\Phi$ coupled
to an infinite set of background higher-spin superfields $H$. Our action
possesses a non-abelian gauge symmetry which naturally generalises the
linearised gauge transformations of conformal half-integer superspin
multiplets. The other fundamental features of this model are: (i)
$\mathcal{S}[\Phi, H]$ is defined on an arbitrary conformally-flat superspace
background; and (ii) the background $\mathcal{N}=1$ superconformal symmetry is
manifest. Making use of $\mathcal{S}[\Phi, H]$, an interacting superconformal
higher-spin theory can be defined as an induced action.
