Turbulence constitutes a nonlinear, extremely high-dimensional, non-conservative complex dynamical system, that exhibits intermittent fluctuations at different scales of motion. Intermittency in turbulence has conventionally been characterized by higher-order moments of velocity gradients (small-scale intermittency) and velocity increments (inertial-range intermittency). In this work, we use data-driven tools rooted in information theory as an improved measure of intermittency in direct numerical simulation data of homogeneous isotropic turbulent flows and turbulent channel flows. Specifically, measures such as Shannon entropy and Kullback-Leibler (KL) divergence, which are functions of the complete probability distribution rather than a few selected moments, are used to quantify turbulence intermittency. First, small-scale intermittency is studied by investigating entropy and KL divergence of velocity gradients and dissipation rates in a broad range of Reynolds numbers. Two distinct regimes appear to exist showing a significant change in intermittency characteristics at a clearly defined transition Reynolds number. This behavior is compared with the variation of higher-order moments of velocity gradients showing anomalous scaling at high Reynolds numbers. Second, we apply information-theoretic measures on velocity increments at different inertial-range scales of the flow. The scaling characteristics are compared with the established scaling laws of velocity structure functions. Finally, these features are studied in channel flow to understand the dependence of intermittency on the large-scale anisotropy of the flow. Overall, this study will derive an improved understanding of turbulence intermittency and its accurate scaling properties.