> It's the role of science and government to find out which anti-biotics, which hormones, and which pesticides case increased rates of which cancers.
Not necessarily.
One of several roles of government is to overcome coordination problems that would otherwise inhibit the production of public goods, such as new antibiotics and other better treatments for bacterial infections.
Another role of government is to regulate markets by creating incentives in the form of laws and legal regulations. In this case, government could estimate the societal cost of agricultural antibiotic use, and then impose taxes on agricultural antibiotic use equal to the amount of that cost. if those costs turn out to be punitive, then tough shit, they'll need to figure it out. The proceeds from tax is levied could then be redirected to basic research as needed.
Of course, this doesn't tend to work so well in practice due to any number of corruption and regulatory capture channels, so it tends to be easier to just ban harmful practices outright, which of course governments also have the power to do.