  We develop a quenched thermodynamic formalism for a wide class of random maps
with non-uniform expansion, where no Markov structure, no uniformly bounded
degree or the existence of some expanding dynamics is required. We prove that
every measurable and fibered $C^1$-potential at high temperature admits a
unique equilibrium state which satisfies a weak Gibbs property, and has
exponential decay of correlations. The arguments combine a functional analytic
approach for the decay of correlations (using Birkhoff cone methods) and
Carath\'eodory-type structures to describe the relative pressure of not
necessary compact invariant sets in random dynamical systems. We establish also
a variational principle for the relative pressure of random dynamical systems.
