If they are, they're terrible at it.
According the the Bureau of Labor Statistics, the Quit Rate for minimum wage-type jobs (leisure and hospitality, retail sales, etc) runs significantly higher than the Quit Rate for average private employment. Indeed, at no point in the 20 years for which we have data (Dec 2000 was the first year collected) were private quits higher than minimum-wage quits.
This data isn't too surprising; minimum wage advocates will sometimes point to high turnover rates and claim that a minimum wage could be an efficiency wage, thus decreasing turnover.* But high turnover rates indicate employers have little (in fact, no) exploitative power. It's hard to exploit workers if they quit.
Additionally, these data undermine the "monopsony" argument for minimum wage. They suggest that minimum wage workers are more mobile than the model requires. What's more, when we see that minimum wage workers do not stay at a minimum wage for long, it suggests that minimum wage employers do indeed face competition for workers: higher-skilled employers!
All data comes from the JOLTS survey.
*There are arguments against the minimum wage as an efficiency wage thesis, but they are not relevant here.
