NORWALK, Conn. — Start in Room 4, just beyond the reception area: A man is having blood drained from a bruised finger. Over in Room 1, a woman is being treated for eye trouble. Next door, in Room 2, a boy is having his throat swabbed.

For more than eight hours a day, seven days a week, 52 weeks a year, an assortment of ailments is on display at the tidy medical clinic on Main Avenue here. But all of the patients have one thing in common: No one is being treated at a traditional doctor’s office or emergency room.

Instead, they have turned to one of the fastest-growing segments of American health care: urgent care, a common category of walk-in clinics with uncommon interest from Wall Street. Once derided as “Doc in a Box” medicine, urgent care has mushroomed into an estimated $14.5 billion business, as investors try to profit from the shifting landscape in health care.

The office here is part of PhysicianOne Urgent Care. Bankrolled by two private investment companies, PhysicianOne has grown into an eight-clinic operation, the largest of its kind in Connecticut, with plans for even greater expansion.