We study a one-dimensional array of bosons with infinite-range interactions mediated by a laser-driven dissipative optical cavity. The cavity-mediated infinite-range interactions open up an alternative pathway to crystallization, hitherto only known for dipolar bosons due to their long-range interactions. In parameter ranges attainable in state-of-the-art experiments, we systematically compare observables for bosons and fermions with infinite-range interactions. At sufficiently large laser pump powers, many observables, including density distributions in real and momentum space, correlation functions, eigenvalues of the one-body density matrix, and superradiance order parameters, become identical for bosons and fermions. We map out the emergence of this cavity-induced crystallization as a function of pump power and contact interactions. We discover that cavity-mediated interactions can compensate a reduction by several orders of magnitude in the strength of the contact interactions needed to trigger crystallization.