Beautiful link Antonio.
I am reading this right, yeah?
"Only 52 natural pesticides have been tested in high-dose animal cancer tests, and about half (27) are rodent carcinogens; these 27 are shown to be present in many common foods."
That means at high enough doses, the compounds manufactured by plants to fend off pests cause cancer in rodent studies. Is that what the sentence means?
Yeah, that's basically it. When a compound is tested for carcinogenicity, it's given at high dose in order to see if there's an effect from it. The problem with this approach is that the compounds may act differently at high dose than at low dose. Specifically, many things that are safe at normal dietary doses are pathogenic at high dose.
To address the OP's question, the only things that I make a point of avoiding in my diet are excess amounts of sugar, PUFAs, and processed meats. Even with these, I don't totally avoid them, just keep them low. I also avoid UV exposure when it's reasonably feasible.