President Obama said Tuesday at the White House that the number of people using to enroll in health insurance is "growing every day." (Associated Press)

President Obama on Tuesday embarked on the difficult task of persuading Americans to reconsider his landmark health-care law after its botched launch, imploring people to sign up as part of a major push to publicize the benefits of the Affordable Care Act.

Republicans immediately lampooned the effort — coming more than three years after passage of the law and two months after its disastrous rollout — and conservative activists vowed to wage a counterassault publicizing the law’s downsides.

But Obama and his Democratic allies expressed confidence that they were finally turning the corner after severe technical issues hobbled, the government Web site where Americans can enroll in health insurance.

“Our poor execution in the first couple months on the Web site clouded the fact that there are a whole bunch of people who stand to benefit,” Obama said at a White House event, flanked by people who said they had benefited from the law. “Now that the Web site’s working for the vast majority of people, we need to make sure that folks refocus on what’s at stake here, which is the capacity for you or your families to be able to have the security of decent health insurance at a reasonable cost.”

The statement marked the start of a new effort by the White House to shine attention on the law. On Sunday, the administration announced that the Web site was finally working well for the “vast majority” of users, although significant bugs remained. The administration had planned a huge push after the law launched on Oct. 1, but those plans were tabled when the site — where Americans could choose among insurance plans in federal marketplaces — barely functioned.

Now, Democratic lawmakers plan to join with the White House in highlighting the benefits of the law, as do a series of outside groups — all with the knowledge that time is short.

Americans must sign up by Dec. 23 to receive coverage that takes effect Jan. 1, and Obama and his team plan to hold daily events highlighting the law until then. The law’s success depends on adding millions of relatively healthy Americans to the insurance rolls — many of whom may have been turned off by the early problems.

Administration officials pointed to traffic on this week as evidence that the site is working much better. On Tuesday, Julie Bataille, communications director for the Centers for Medicare and Medicaid Services, said that 1.38 million people had visited the Web site from Monday through noon Tuesday.

Bataille said that 16,000 people on Monday were placed in an online queue, a new feature of the system that notifies them if the site is too crowded and sends them an e-mail inviting them to try again when it is less clogged.

The queuing began when the site had about 15,000 fewer users than the administration’s target capacity of 50,000. About 60 percent of the people who were sent to the queue returned to the site later in the day, she said.

Still, it was clear that bugs remained in the system. The National Association of Health Underwriters, which represents more than 100,000 licensed health-insurance agents and brokers, on Tuesday asked the Obama to fix the Web site’s “back end” technical obstacles.

These problems — separate from troubles consumers are experiencing — are preventing brokers from enrolling hundreds of thousands of people in coverage, according to a letter from Janet Trautwein, the association’s chief executive.

Jennifer Palmieri, White House communications director, said: “It’s not that we think the Web site is fixed and done. At the same time, we need to be talking about the benefits of the law. That’s the theory of what we need to do. It’s helpful for Democrats.”

Conservatives pledged a robust response to the White House-led push. Before Obama spoke, Senate Minority Leader Mitch McConnell (R-Ky.) bemoaned another “campaign-style event” that he said “won’t solve the myriad problems facing consumers under Obamacare.”

Americans for Prosperity, a group funded by billionaire conservatives David and Charles Koch, has spent $27 million over the past few months highlighting what it views as flaws in the legislation. The group plans to spend more on television and radio ads, online campaigns and other efforts to disparage the law in coming months.

Tim Phillips, the group’s president, said the efforts will highlight problems with the law generally and will also target specific Democrats from conservative states who supported it, including Sens. Mark Begich (D-Alaska) and Mary Landrieu (D-La.).

“If the president rolls out individual elements that are supposedly a good thing and he wants to highlight, we’re absolutely going to continue rolling out new problems of Obamacare,” Phillips said, adding that the focus would be on Americans who lose access to their preferred set of doctors or specialists.

But the president and his allies expressed confidence they would prevail.

“We’ve learned not to make wild promises about how perfectly smooth it’s going to be at all times,” Obama said. “But if you really want health insurance through the marketplaces, you’re going to be able to get on and find the information that you need for your families.”

Obama framed his remarks as a reminder of why he says he pursued passage of the Affordable Care Act in the first place.

“For too long, few things left working families more vulnerable to the anxieties and insecurities of today’s economy than a broken health-care system,” Obama said. “So we took up the fight because we believe that, in America, nobody should have to worry about going broke just because somebody in their family or they got sick.”

A range of outside groups is helping the White House in its publicity effort. Enroll America, which was founded to support the rollout of the Affordable Care Act, is launching a “Coverage is Coming” campaign featuring more than 1,000 events nationwide this month.

Former senior White House official Anne Filipic, who leads the group, said Enroll America will also begin following up in person, via e-mail and through phone calls with roughly 50,000 people who identify themselves as underinsured or uninsured.

Even with the Web site’s problems, Filipic said, “it hasn’t in any way felt like a death blow to our efforts,” because most people were not ready to enroll before now.

People such as Tracy Young, a 64-year-old self-employed Tucson resident who helps disabled adults find jobs, will be crucial to the administration’s hopes for the law. Young, whose individual insurance rates have jumped, was able to use in mid-October after switching her Internet browser.

After reviewing 119 plans, Young chose one in which she will pay $399 a month — about $50 less than her current policy — with better benefits.

“Within two weeks I had an invoice and a group number and a subscriber number,” Young said Tuesday. “I sent in my check at the end of November. I did see the check cleared, and I should be getting my insurance card. So it worked for me. I was thrilled.”

Amy Goldstein and Lena H. Sun contributed to this report.