UPS said it was cutting its business with Amazon in half by the middle of next year because of profitability concerns. Shares plunged.

Amazon is the king of fast delivery—it pioneered the two-day shipping model customers have come to expect. But to stay ahead of Target and Walmart, it’s overhauling its distribution network.

Amazon has spent years building its logistics services, including its warehouses and its aircraft fleet. The company has used those resources to offer its own shipping options. Late last year ...