I would encourage you to read Antibiotic Growth-Promoters in Food Animals ( http://www.fao.org/docrep/ARTICLE/AGRIPPA/555_EN.HTM ) (the fao domain is the Food and Agriculture Organization of the United Nations) or the introduction of Antimicrobial Growth Promoters Used in Animal Feed: Effects of Less Well Known Antibiotics on Gram-Positive Bacteria ( https://www.ncbi.nlm.nih.gov/pmc/articles/PMC153145/ )

The Wikipedia article https://en.wikipedia.org/wiki/Antibiotic_use_in_livestock also covers it:

> Certain antibiotics, when given in low, sub-therapeutic doses, are known to improve feed conversion efficiency (more output, such as muscle or milk, for a given amount of feed) and/or may promote greater growth, most likely by affecting gut flora.[15] However, any antibiotics deemed medically important to humans by the CDC are illegal to use as growth promoters in the U.S. Only drugs that have no association with human medicine – and therefore no risk to humans – are allowed to be used for this purpose.[5][16] It is also important to note that some drugs listed below are ionophores, which are not antibiotics and do not pose any potential risk to human health.

They are indeed not growth hormones, but they do change the gut bacteria balance and make it easier to extract calories from the feed stock. This has nothing to do with actually being sick or not.