Kinetically constrained models (KCMs) were introduced by physicists to model the liquid-glass transition. They are interacting particle systems on Z^d in which each element of Z^d can be in state 0 or 1 and tries to update its state to 0 at rate q and to 1 at rate 1 − q, provided that a constraint is satisfied. In this article, we prove the first non-perturbative result of convergence to equilibrium for KCMs with general constraints: for any KCM in the class termed "supercritical" in dimension 1 and 2, when the initial configuration has product Bernoulli(1 − q') law with q' different from q, the dynamics converges to equilibrium with exponential speed when q is close enough to 1, which corresponds to the high temperature regime.