We develop a manifestly conformal approach to describe linearised (super)conformal higher-spin gauge theories in arbitrary conformally flat backgrounds in three and four spacetime dimensions. Closed-form expressions in terms of gauge prepotentials are given for gauge-invariant higher-spin (super) Cotton and (super) Weyl tensors in three and four dimensions, respectively. The higher-spin (super) Weyl tensors are shown to be conformal primary (super)fields in arbitrary conformal (super)gravity backgrounds, however they are gauge invariant only if the background (super) Weyl tensor vanishes. The proposed higher-spin actions are (super) Weyl-invariant on arbitrary curved backgrounds, however the appropriate higher-spin gauge invariance holds only in the conformally flat case. We also describe conformal models for generalised gauge fields that are used to describe partially massless dynamics in three and four dimensions. In particular, generalised higher-spin Cotton and Weyl tensors are introduced.