Internet-only polls fared well in 538's analysis of 2012 election polling: http://fivethirtyeight.blogs.nytimes.com/2012/11/10/which-po...
crowd-sourcing platforms need their own bias adjustments and I'm not even convinced that it's possible to adjust for the bias caused by using these platforms, unless you really are interested in putting a lower bound on human ability. Not a statement about the quality of people on Turk, just the amount of effort they often (rightfully!) put into below-developed-world-minimum-wage work...
There's a big difference b/w posting flyers/ads in target-rich environments and providing a low friction way for people to get involved in the study. But if you do something on Amazon Turk you're basically guaranteeing a very particular type of person, because random guy with a high paying job probably isn't going to have a turk account and won't have the patience to create one just for your survey.
In some cases you can make a good case that this doesn't totally invalidate any sort of generality in the results, but for the most part I throw out papers -- on face -- that proceed in this ways without extended, evidence-based discussion of why the hell I can expect any sort of conclusion except "our result generalizes to a population of people who really need an extra dollar or two (literally) and are looking for pretty much any way to get it."
Online surveys are certainly becoming more popular as they are significantly cheaper to conduct than the alternatives, and yield publishable results that garner media attention. There are peer reviewers that will be sympathetic to these issues, regardless of the method's robustness.
However, there are others that would say this reeks of dredging (p-hacking) in a very murky pool of data. Their "scepticism" rarely makes the New York Times (or a bestselling book), though.