We present an extensive study of the enzyme-mediated, isothermal formation of covalently cross-linked gelatin gels. We find that the enzymatic activity in the forming network is drastically reduced compared with that in solution, and show that this can be attributed to the growing level of cross-link induced geometric constraints which impede translational and rotational motions. Thanks to the slowness of these kinetics, we monitor the concomitant build-up of the shear modulus G' and of the optical turbidity T, which indicates that gelation is associated with the development of a high level of inhomogeneity. We find that, as the gelatin concentration c(G) is varied, the levels of G' and T are strongly anti-correlated. Moreover, the lower cG, the more precocious the emergence of T. We are able to analyze inhomogeneity development in terms of the amplification of structural fluctuations via the coupling between the kinetics of the cross-linking reaction and the osmotic flow driven by swelling pressure fluctuations. We expect this positive feedback mechanism to be efficient in any slow, irreversible gelation process.