Conformable derivatives have deep connections with fractals and local fractional calculus. This paper presents a generalized concept of higher-order fuzzy conformable differentiability for fuzzy-valued functions and interprets higher-order fuzzy conformable differential equations using this concept. We propose the fuzzy conformable Laplace transform (FCLT) as an alternative to the fuzzy Sumudu integral transform, and fuzzy Shehu transform method; and establish some potential properties of FCLT, related to higher-order fuzzy conformable derivatives. The FCLT method is very convenient for solving fuzzy conformable linear higher-order differential equations. We illustrate linear second-order and linear fourth-order fuzzy conformable initial value problems having at most four and sixteen different solutions, respectively.