In this paper, we study a class of quadratic-exponential growth BSDEs with jumps. The quadratic structure was introduced by Barrieu & El Karoui (2013) and yields a very useful universal bound on the possible solutions. With the bounded terminal condition as well as an additional local Lipschitz continuity, we give a simple and streamlined proof for the existence and the uniqueness of the solution. The universal bound and the stability result for the locally Lipschitz BSDEs with coefficients in the BMO space enable us to show the strong convergence of a sequence of globally Lipschitz BSDEs. The result is then used to generalize the existing results on the Malliavin’s differentiability of the quadratic BSDEs in the diffusion setup to the quadratic-exponential growth BSDEs with jumps.