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Over the past decade, the online video industry has greatly expanded the volume of visual data that is streamed and shared over the Internet. Moreover, because of the increasing ease of video capture, many millions of consumers create and upload large volumes of User-Generated-Content (UGC) videos. Unlike streaming television or cinematic content produced by professional videographers and cinemagraphers, UGC videos are most commonly captured by naive users having limited skills and imperfect technique, and often are afflicted by highly diverse and mixed in-capture distortions. These UGC videos are then often uploaded for sharing onto cloud servers, where they further compressed for storage and transmission. Our paper tackles the highly practical problem of predicting the quality of compressed videos (perhaps during the process of compression, to help guide it), with only (possibly severely) distorted UGC videos as references. To address this problem, we have developed a novel Video Quality Assessment (VQA) framework that we call 1stepVQA (to distinguish it from two-step methods that we discuss). 1stepVQA overcomes limitations of Full-Reference, Reduced-Reference and No-Reference VQA models by exploiting the statistical regularities of both natural videos and distorted videos. We show that 1stepVQA is able to more accurately predict the quality of compressed videos, given imperfect reference videos. We also describe a new dedicated video database which includes (typically distorted) UGC reference videos, and a large number of compressed versions of them. We show that the 1stepVQA model outperforms other VQA models in this scenario. We are providing the dedicated new database free of charge at https://live.ece.utexas.edu/research/onestep/index.html