How to effectively fuse temporal information from consecutive frames remains to be a non-trivial problem in video super-resolution (SR), since most existing fusion strategies (direct fusion, slow fusion, or 3D convolution) either fail to make full use of temporal information or cost too much calculation. To this end, we propose a novel progressive fusion network for video SR, in which frames are processed in a way of progressive separation and fusion for the thorough utilization of spatio-temporal information. We particularly incorporate multi-scale structure and hybrid convolutions into the network to capture a wide range of dependencies. We further propose a non-local operation to extract long-range spatio-temporal correlations directly, taking place of traditional motion estimation and motion compensation (ME&MC). This design relieves the complicated ME&MC algorithms, but enjoys better performance than various ME&MC schemes. Finally, we improve generative adversarial training for video SR to avoid temporal artifacts such as flickering and ghosting. In particular, we propose a frame variation loss with a single-sequence training method to generate more realistic and temporally consistent videos. Extensive experiments on public datasets show the superiority of our method over state-of-the-art methods in terms of performance and complexity. Our code is available at https://github.com/psychopa4/MSHPFNL.