The accident happened in the early evening at the seaside town of Berck-sur-Mer where hundreds of thousands of people gather each year for the festival.
Four women, including two pensioners aged 75 and 82, are being treated for life-threatening head injuries in intensive care, the emergency services and local officials said.
The driver, who was lightly injured in the accident, faces a possible charge of causing injuries unintentionally.
Popular with families and drawing visitors from around the world, the annual Berck-sur-Mer festival is held over several days, featuring acrobatic kite flying shows and build-your-own workshops.
One of the 11 injured was a four-year-old girl who suffered shock after witnessing the accident, an official said.