Novel view synthesis of dynamic scenes is becoming important in various applications, including augmented and virtual reality.We propose a novel 4D Gaussian Splatting (4DGS) algorithm for dynamic scenes from casually recorded monocular videos. To overcome the overfitting problem of existing work for these real-world videos, we introduce an uncertainty-aware regularization that identifies uncertain regions with few observations and selectively imposes additional priors based on diffusion models and depth smoothness on such regions.This approach improves both the performance of novel view synthesis and the quality of training image reconstruction. We also identify the initialization problem of 4DGS in fast-moving dynamic regions, where the Structure from Motion (SfM) algorithm fails to provide reliable 3D landmarks. To initialize Gaussian primitives in such regions, we present a dynamic region densification method using the estimated depth maps and scene flow. Our experiments show that the proposed method improves the performance of 4DGS reconstruction from a video captured by a handheld monocular camera and also exhibits promising results in few-shot static scene reconstruction.