We study conductance fluctuations in a two-dimensional electron gas as a function of chemical potential (or gate voltage) from the strongly insulating to the metallic regime. Power spectra of the fluctuations decay with two distinct exponents (1/v_l and 1/v_h). For conductivity $sigmasim 0.1 e^{2}/h$, we find a third exponent (1/v_i) in the shortest samples, and non-monotonic dependence of v_i and v_l on sigma. We study the dependence of v_i, v_l, v_h, and the variances of corresponding fluctuations on sigma, sample size, and temperature. The anomalies near $sigmasimeq 0.1 e^{2}/h$ indicate that the dielectric response and screening length are critically behaved, i.e. that Coulomb correlations dominate the physics.
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