Turbulence mitigation refers to the stabilization of videos with nonuniform deformations due to the influence of optical turbulence. Typical approaches for turbulence mitigation follow averaging or dewarping techniques. Although these methods can reduce the turbulence, they distort the independently moving objects, which can often be of great interest. In this paper, we address the novel problem of simultaneous turbulence mitigation and moving object detection. We propose a novel three-term low-rank matrix decomposition approach in which we decompose the turbulence sequence into three components: the background, the turbulence, and the object. We simplify this extremely difficult problem into a minimization of nuclear norm, Frobenius norm, and 21 norm. Our method is based on two observations: First, the turbulence causes dense and Gaussian noise and therefore can be captured by Frobenius norm, while the moving objects are sparse and thus can be captured by 21 norm. Second, since the object's motion is linear and intrinsically different from the Gaussian-like turbulence, a Gaussian-based turbulence model can be employed to enforce an additional constraint on the search space of the minimization. We demonstrate the robustness of our approach on challenging sequences which are significantly distorted with atmospheric turbulence and include extremely tiny moving objects.