G$_{2}$-continuous trajectories are crucial for vehicles under non-holonomic dynamic constraints to run smoothly. In this paper, we present a path smoothing method based on piece-wise quadratic Bézier curves, which can fit a set of sequential 2D collision-free path points generated by a higher-level path planner. The generated trajectories are G$_{2}$-continuous except at the inflection points, and points of local curvature maxima appear exactly at path points and nowhere else. The local curvature control ensures that the trajectories keep high fidelity with the original paths, thus maintaining key properties of the original paths, such as free collision and gentle turns. Furthermore, we develop a joint optimization method for both fidelity and continuity, which has an efficient analytical solution for each iteration. Extensive experiments in both simulated and real scenarios validate that 1) the smoothed trajectories can deviate less than 10 cm from the original paths, 2) the average curvature differences at junctures are under 1e-4 (1/m), 3) the smoothing time reduced by about 50% versus high-order bézier methods, showing superior performance of the proposed method in terms of fidelity, continuity, efficiency, and practicability.