With Monday’s deadline for enrollment in 2014 approaching, Obamacare—sorry, former Speaker Pelosi, but that’s what it’s come to be known as—presents something of a paradox. After the disastrous problems early on with the federal online insurance market, HealthCare.gov, the sign-up process has gone much better than could have been expected a few months ago. But opinion polls show that a majority of Americans still oppose the health-care law. And many Democrats fear that, come November, it could cost them the Senate.

What is going on? First, the numbers. According to the latest figures from the White House, more than six million people have signed up for private insurance policies offered under the Affordable Care Act. Another three or four million low-income families have enrolled in the expanded Medicaid program, and, under a provision of the law introduced in 2010, about the same number of young people have been added to their parents’ private insurance plans. By the Monday deadline, the figures will likely be higher. On Thursday, 1.5 million people visited HealthCare.gov, and more than four hundred thousand people placed calls to the insurance exchange, the Department of Health and Human Services said in a tweet.

Of course, these figures can be parsed in numerous ways. A good number of the enrollees, perhaps fifteen or twenty per cent, haven’t paid their first monthly premiums yet, and we can’t be absolutely sure that they will. The churn rate—the proportion of people who drop out after a few months—hasn’t been published. Among those signing up, we don’t know what the final age distribution will be. (In the three months to the end of March, seventy per cent of the enrollees were thirty-five or older, which is more than the government was hoping for.) We don’t know how many of the enrollees were previously uninsured and how many were switching plans—either voluntarily or because their old ones were cancelled.

Despite these qualifications, though, the underlying message is a positive one. Lots of Americans who previously couldn’t obtain insurance, either because they couldn’t afford it or because insurers wouldn’t offer it to them, now have health coverage. Despite all the negative publicity surrounding HealthCare.gov’s launch, despite all the attacks by Republicans and other opponents of the law, despite all the carve-outs and delayed deadlines imposed by an anxious White House, Obamacare is up and running. Contrary to the predictions of doom, it hasn’t collapsed under the weight of its own complications.

As was the case with the Massachusetts reform on which it was largely based, it will take a couple of years, or even longer, before we can really judge how the new system is working. Because each state is its own insurance market, the outcomes will differ widely across the country. In big Democrat-run states that are running their own insurance exchanges, such as California and New York, and in states like Illinois and Michigan that are operating exchanges in partnership with the federal government, the prognosis is pretty good. Large numbers of people are signing up, and there’s quite a bit of competition among insurers. Over time, as people become increasingly familiar with the system, enrollments are likely to ramp up further, which is what the designers of the plan envisaged.

In hard-line Republican states such as Texas and Louisiana, which have leaders who are implacably opposed to the law, the outlook is much less favorable. Enrollment rates on HealthCare.gov are much lower than elsewhere, which isn’t surprising given that some of these states, in a blatant effort to thwart the reform, have placed strict restrictions on who can serve as counsellors to guide people through the enrollment process. And low-income families in these states, since their governors have refused to coöperate with the reform, are denied the option of signing up for the expanded Medicaid program. In areas like these, there is a possibility of a negative spiral, with low rates of enrollment prompting insurers to raise rates, which, in turn, dissuades more people from enrolling.

An objective reading of this record would be that Obamacare, after a horrible foul-up in rolling out HealthCare.gov, has made a good deal of progress, and some of its problems are the consequence of deliberate efforts to undermine it.

Since the government is effectively handing out money to families of modest means—figures from the Department of Health and Human Services show that about four in five of the enrollees are eligible for a federal subsidy—it is perhaps not so surprising that the sign-up figures have jumped sharply over the past few weeks. After months of bad news, it’s still something for the Obama Administration, and all supporters of health-care reform, to cheer. So far, though, this turnaround isn’t really reflected in the opinion polls. As the glitches on HealthCare.gov have been fixed, the level of opposition to the reform has fallen back from the levels it reached in November and December: that was only to be expected. Taking a longer view, though, the striking thing is how little public attitudes have changed.

The results in individual polls bounce around a bit, and there are some shocking outliers. On Friday, for instance, a new survey from AP-GfK suggested that just twenty-six per cent of Americans support the health-care reforms. But the Real Clear Politics poll of polls, which averages the individual surveys, shows a remarkable degree of stability. On December 15, 2009, about three months before Obama signed the Affordable Care Act, 52.7 per cent of poll respondents opposed it, and 37.8 per cent supported it. On March 25, 2014, 53.2 per cent opposed it, and 39.8 per cent supported it.

These figures are discouraging for supporters of the reform. And yet Americans’ views of the A.C.A. aren’t as monolithic as they might appear. For one thing, the polls show consistently that most people favor keeping the reform and amending it, rather than repealing it, which is still the Republican position. That explains why, this week, six moderate Democrats who are up for election in November announced their intention to modify the A.C.A. in various ways.

Even more striking, many of the individual elements of the reform are highly popular, and have been all along. The latest tracking poll from the Kaiser Family Foundation found that eighty per cent of respondents are in favor of allowing young adults to stay on their parents’ insurance plans; seventy-nine per cent favor eliminating the “doughnut hole” in Medicare’s prescription drugs plan; seventy-seven per cent favor providing subsidies to folks who can’t afford coverage; seventy-four per cent favor the expansion of Medicaid; and seventy per cent favor preventing insurance companies from dropping people with preëxisting conditions. The Medicare tax increase on upper-income households is less popular, but it still has majority support: fifty-six per cent. About the only aspect of the plan that evoked a negative reaction was the individual mandate, which just thirty-five per cent of respondents viewed favorably.

Why, then, does the over-all reform remain so unpopular? Part of the answer is that it faces the same challenges as any other progressive program designed primarily to help a relatively small section of the population—in this case, the uninsured. Lots of people who don’t directly benefit, at least for the moment, because they already have insurance, don’t see much in it for them. But there is also a big communications problem, which still hasn’t been resolved. The Republicans, with the enthusiastic help of conservative media outlets, have done a fiendishly effective job of stigmatizing what is essentially a private-sector solution as a big-government intrusion, and a handout to the unworthy.

Back to that poll from the Kaiser Family Foundation. It suggests that one in three Americans believe that the A.C.A. established a “death panel” to make decisions about the end-of-life coverage that people receive under Medicare, and another one in four Americans aren’t sure whether such a panel exists. (It doesn’t.) On top of that, nearly half of Americans think that the A.C.A. allows undocumented immigrants to receive federal subsidies for the purchase of insurance on the new exchanges. This despite the fact that the law explicitly denies subsidies to the undocumented, and even bars them from purchasing plans through the exchanges at full cost.

A couple of weeks ago, I suggested that it was time for more Democrats to abandon their caution and embrace Obamacare as a historic reform that makes health care more accessible and affordable. A good way to start would be by making more effort to dispel some of the damaging myths that still surround the program.

Photograph by Andrew Harrer/Bloomberg/Getty.