The replication crisis that's plaguing much of the social sciences, but especially psychology, did not cherry pick studies. It started with an effort to replicate studies
only from high impact well regarded journals in psychology. [1] It found that 64% of the studies could not be replicated, leading to the curious outcome that if you assumed the literal and exact opposite of what you read in psychology (e.g. - what is said to be statistically significant, is not) - you would tend to be substantially more accurately informed than those who believe the 'science.' [1]
But more to our discussion, two of the journals from which studies were chosen were Psychological Science - impact factor 6.128, and the Journal of Personality and Social Psychology - impact factor 5.733. The replication success rate for those journals was 38% and 23% respectively. I'm certain you know, but impact factor is the yearly average number of citations for each article published in a journal. A high impact factor is generally anything above about 2. These are among the crème de la crème of psychology, and they're worthless.
As you mention pubmed, preclinical research is also a field with just an absolutely abysmal replication rate. And once again these are not cherry picked. In an internal replication study Amgen, one of the world's largest biotech companies, alongside researchers from MD Anderson, one of the world's premier cancer hospitals, were only able to replicate 11% of landmark hematology and oncology papers. [2] Needless to say those papers, and their now unsupported conclusions, were acted upon in some cases.
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All that said I do completely agree with you that the current system of publish or perish is playing into this, but your characterization of the current state of bad science is inaccurate. Bad science is becoming ubiquitous. However, I'm not as optimistic that there is any clean solution. There are currently about 400 players in the NBA. If you increased that 4,000 what would you expect to happen to the mean quality and the lowest common denominators? Suddenly somebody who would normally not even make it into the NBA is a first round pick. And science is a skill like any other that relies on outliers to drive it forward. We now have a system that's mostly just shoveling people through it and outputting 'scientists' for commercial gain. The output of this system is, in my opinion, fundamentally harming the entire system of science and education. And this is a downward spiral because now these individuals of overall lower quality are working as the mentors and advisers for the next generation of scientists, and actively 'educating' the current generation of doe-eyed students. This is something that will get worse, not better, over time.
[1] - https://en.wikipedia.org/wiki/Replication_crisis#Psychology_...
[2] - https://www.taconic.com/taconic-insights/quality/replication...
[3] - http://graphics8.nytimes.com/packages/pdf/education/harvarde...