See for example this, this, or this - there are safe organic pesticides, but the most effective ones tend to be in the "shockingly lethal products of evolution" category, and are only believed to be safe because of vitalism myths.

The less nutrients thing refers to crops like golden rice, which has been genetically engineered to contain Vitamin A to help stave off 1-2 million people dying every year from deficiency. Naturally, organic food activists are violently opposed to it(sometimes literally).

The more damage refers mostly to the Borlaug hypothesis - using more efficient scientific farming techniques means you can grow more food on less land, which means we need to do less deforestation and can leave more land in its natural state. Also, remember that prices contain information - farmers need to pay for all the resources they use, and those costs are embedded in the price of the food. If organic food costs twice as much, it's probably because organic farming uses twice as many resources. (Admittedly, this does include resources like labour, which isn't strictly a "green" concern, but I like people, so I care about how hard they have to work on top of environmental issues)