This deceptively simple question is difficult to prove. In a widely-read science blog post, Dr. Christie Wilcox (her PhD is in marine biology, not agriculture) argues that the supposed benefits of organic farming are all myths.
While some of her specific claims, such as that organic farmers are allowed to use the incredibly toxic natural compound rotenone, are off-mark, much of her critique appears to stand. The reasons are various, but telling: there are costs and benefits to different agricultural systems, and organic farms face many of the same challenges of conventional farming. For example, herbicides allow no-till farming, with myriad benefits for soil structure and water quality. It is extremely difficult to practice no-till without some means of removing weeds.
Certified organic farms may not use synthetic pesticides, herbicides, fungicides, etc. But if organic farmers choose to use a natural compound to kill weeds or pests there is nothing to guarantee that such a compound would be less toxic to the ecosystem and to humans. Natural compounds on organic farms may be more toxic than their synthetic analogues!
Unfortunately, while there are comprehensive databases of the type and amount of pesticides found on conventional food (see the Environmental Working Group's annual Dirty Dozen list), no such testing of natural compounds on organic produce is conducted. Apparently synthetic compounds are investigated more than natural compounds, perhaps due to an unstated belief in the Naturalistic Fallacy, i.e. that natural chemicals must be healthier than synthetic chemicals. But many plants (for example, nightshades) contain totally natural compounds... that are totally toxic as well.
A good take-home message would be to wash all of your fruit and veggies, whether they are organic or not. And learn which plants are poisonous. I'm not even going to get into all of the (natural and synthetic) chemicals and preservatives in grains, let alone refined flours, etc.