Where I usually break down and take the car, is leafy greens. I always make sure I have a plastic bag big enough to cover them completely (or use two), but if it's more than a bag of baby lettuces, they just get beaten to bits in a pannier. Especially if it's anything larger like chard or beets with greens, that sticks half out of the pannier and flaps around in the wind.

What I wind up doing is taking the car to the winter CSA pickup, which I do midweek and is usually heavy on greens, then supplement that with a ride to the market on Saturday for juice, eggs, strawberries, meat, fish and dairy products, and any extra veggies I might need. With the small soft cooler and freezer packs for the fish and frozen meat, it's usually all I can fit in my bags anyway.