With a network of 35,000 doctors and 72 hospitals, Oscar offers insurance in New York City, Long Island, Rockland and Westchester counties, with monthly premiums ranging from $227 to $603 for an individual depending on coverage levels, deductibles and regions.

Its telemedicine feature, using a company called Teladoc, is intended to give subscribers phone or online consults within 30 minutes with New York-based primary care practitioners who can access their medical records.

"If your baby's crying at 10 o'clock at night and want someone to talk through the issue with, you utilize it. It's not a replacement for your ongoing primary care relationship," Nazemi said. "We modeled it such that you can engage as many times as you want. It's an unlimited feature."

At nonprofit New York startup Health Republic Insurance, chief executive Debra Friedman said phone volume rose 27 percent in one week. She predicts "rapid" enrollment expansion over the next two weeks, approaching the Dec. 23 deadline for coverage starting immediately on Jan. 1, though open enrollments continue through March for 2014 coverage at rates approved by the state Department of Financial Services in July.

Health Republic, with a network of 70,000 providers and more than 135 hospitals statewide, offers plans in 32 counties for individuals and small businesses. Like Oscar, it arranged providers through the MagnaCare network that many self-insured companies use. Its premiums range from $150 to $523 monthly for an individual and from $237 to $570 in small group plans.

"Membership enrollment to date and market share of individuals in the exchange, as well as phone volumes and website traffic, are up dramatically week over week," Friedman said.

She declined to disclose total enrollment so far at the Co-Op, or consumer-operated and oriented plan, but said the exchange lets consumers compare prices for comparable products and Health Republic's are among the lowest. "We're well exceeding our market share," she said.

A third startup in the exchange, Northshore-LIJ CareConnect Insurance, covers care through its affiliated system of 16 hospitals in Long Island, Queens, Staten Island and Manhattan and more than 4,500 physicians and other providers. Its premiums range from $183 to $568 for an individual and from $467 to $747 in its small group plans.

President Alan Murray said Thursday the numbers have grown rapidly and he expects that to continue the next few weeks, including small businesses. He declined to disclose enrollment so far except to say it's in four digits.

"We're going to expect a spike in the next few weeks as we get close to the end of the year. I think that will spill over into next year," Murray said.

As part of an integrated system that includes hospitals and doctors, he said Northshore-LIJ differs from the other insurers in New York and the market nationally.

"In the incredibly confusing world of health care, we have the ability to make it very, very simple. … It's all inside our company," he said.

Many uninsured applicants are expected to qualify for income-based tax credits. Nazemi and Friedman noted significant interest in so-called silver plans, the second-highest coverage levels, where premiums could be offset by those credits.

Consumers can also shop the networks for particular doctors or hospitals associated with a plan. The Health Department said it has reviewed each to ensure it meets the same adequacy standards required of health maintenance organizations in New York, while the state also requires covering out-of-network care with the same out-of-pocket cost if a member needs a provider unavailable in the network. That would apply, for example, to breast reconstruction surgery that nobody in the network performs.