We give a method to construct non-trivial $p$-harmonic morphisms via conformal change of the metric on the domain and/or the target manifold (Theorems 2.1, 2.5 and 2.8). As applications, we show the existence of higher dimensional harmonic spheres in general manifolds (Theorem 2.10) generalizing Sacks and Uhlenbeck's result on harmonic 2-spheres, prove some existence theorems for non-trivial $p$-harmonic morphisms between conformally flat spaces (Theorems 2.12, 3.1 and 3.2), give a method to construct minimal foliations via $p$-harmonic morphisms and show that many $\mathbf{R}^{m}$ with a conformally flat metric admit minimal foliations of codimension greater than two (Theorems 3.3, 3.4).