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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 $\mathcal{S}[φ,h]$ describing a complex scalar field $φ$ coupled to an infinite set of background CHS fields $h$, with $\mathcal{S}[φ,h]$ possessing a non-abelian gauge symmetry. Two characteristic features of the perturbative constructions of $\mathcal{S}[φ, h]$ 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) $\mathcal{S}[φ, h]$ 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 $\mathcal{S}[Φ, H]$ for a conformal scalar/chiral multiplet $Φ$ 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) $\mathcal{S}[Φ, H]$ is defined on an arbitrary conformally-flat superspace background; and (ii) the background $\mathcal{N}=1$ superconformal symmetry is manifest. Making use of $\mathcal{S}[Φ, H]$, an interacting superconformal higher-spin theory can be defined as an induced action.