ÉVIAN-LES-BAINS, France—Bottled water, which recently dethroned soda as America’s most popular beverage, is facing a crisis.

A consumer backlash against disposable plastic plus new government mandates and bans in places such as zoos and department stores have the world’s biggest bottled-water makers scrambling to find alternatives.

Evian this year pledged to make all its plastic bottles entirely from recycled plastic by 2025, up from 30% today and among the boldest goals in the industry. Executives at parent company Danone SA hope the move will help it regain market share and win over plastic detractors who are already pressuring the makers of straws, bags and coffee cups.

There’s a big problem. The industry has tried and failed for years to make a better bottle.

Existing recycling technology needs clean, clear plastic to make new water bottles, and bottled-water companies say low recycling rates and a lack of infrastructure have stymied supply. Danone, for its part, is betting the reputation of its flagship water brand on a new technology that claims to turn old plastic from things like dirty carpets and sticky ketchup bottles into plastic suitable for new water bottles.