Causal Graph Discovery (CGD) is the process of estimating the underlying probabilistic graphical model that represents joint distribution of features of a dataset. CGD-algorithms are broadly classified into two categories: (i) Constraint-based algorithms (outcome depends on conditional independence (CI) tests), (ii) Score-based algorithms (outcome depends on optimized score-function). Since, sensitive features of observational data is prone to privacy-leakage, Differential Privacy (DP) has been adopted to ensure user privacy in CGD. Adding same amount of noise in this sequential-natured estimation process affects the predictive performance of the algorithms. As initial CI tests in constraint-based algorithms and later iterations of the optimization process of score-based algorithms are crucial, they need to be more accurate, less noisy. Based on this key observation, we present CURATE (CaUsal gRaph AdapTivE privacy), a DP-CGD framework with adaptive privacy budgeting. In contrast to existing DP-CGD algorithms with uniform privacy budgeting across all iterations, CURATE allows adaptive privacy budgeting by minimizing error probability (for constraint-based), maximizing iterations of the optimization problem (for score-based) while keeping the cumulative leakage bounded. To validate our framework, we present a comprehensive set of experiments on several datasets and show that CURATE achieves higher utility compared to existing DP-CGD algorithms with less privacy-leakage.