The premise is logical, at least for organic shoppers, if only adjacently valid scientifically. Poor diet has been correlated to increased cancer risk. And pesticides have been correlated to cancer risk. Basically, they see cancer and pesticides as a surrogate marker for each other so if there are fewer pesticides, it would mean less cancer. Unfortunately, rational people know that organic food actually does not have fewer pesticides, it instead uses toxic poison that can be found in nature. Studies have instead shown that organic food uses more pesticides than conventional food.
Outside epidemiological advocacy the organic food-pesticide-cancer link doesn't pass the smell test. There are spray and count activists dumping frogs in buckets of pesticides and then waiting to see how many get cancer, of course, but actual science doesn't work that way. Still, population studies have their place, especially when looking at a mass populace. If organic food reduces cancer risk, a giant population of organic consumers should get less cancer over the long term.
Sorry, organic food shoppers, your mice could end up looking like this no matter what they eat. Link: Forbes
A recent paper follow 623,080 middle-aged UK women who reported their consumption of organic food over a period of 9.3 years. That doesn't sound like a long time? In Rachel Carson's "Silent Spring", she claimed people sprayed DDT and got cancer and died a few months later so 9 years is plenty. 53,769 cases showed no meaningful difference in cancer incidence between organic food and normal food consumers. Organic food consumers even showed a small increased risk of breast cancer while they showed a reduction in the risk of non-Hodgkin lymphoma.
Only activists are going to highlight those differences as meaningful (and those selling them books - David Perlmutter's claim that giving up wheat can prevent half of Alzheimer's cases was based on a possible association in 13 people) but there are lots of other variables that could account for it. So buy food you like that tastes the way you want and ignore marketing hype on the label.
Citation: K E Bradbury, A Balkwill, E A Spencer, A W Roddam, G K Reeves, J Green, T J Key, V Beral, K Pirie and The Million Women Study Collaborators, 'Organic food consumption and the incidence of cancer in a large prospective study of women in the United Kingdom', British Journal of Cancer , 27 March 2014 doi:10.1038/bjc.2014.148