ROME — After months of supply shortages and embarrassing blunders, Europe's coronavirus vaccine campaign is at last sprinting ahead, renewing hopes that the continent might meet its initial inoculation goals and tame the virus even while relaxing restrictions.

Across the European Union, countries are touting new daily vaccination records. Supply concerns have eased. The E.U. is now administering roughly the same number of daily per capita doses as the United States, trending up while America trends down.

Some European countries have helped their cause by sorting out logistical problems or enlisting family doctors and pharmacists to administer shots. But the bloc, more broadly, owes its turnaround to a deepened reliance on Pfizer-BioNTech, which has ramped up vaccine production — beyond what was initially expected — as other options have faltered or come too slowly.

Mostly using Pfizer, France’s pace of daily doses has increased 60 percent over the past month. Italy has accelerated by 90 percent, Germany by 145 percent.

While the United States has doses widely available to any adults who want them — and is struggling to find takers — many people in the E.U. are not yet eligible. But appointments are starting to open up in Europe, even for younger cohorts.

E.U. officials, noting President Biden’s target of getting at least one dose to 70 percent of U.S. adults by July 4, say that by midsummer Europeans and Americans will be in roughly the same situation.

“The U.S. has a similar goal, and this shows how much our vaccination campaigns are aligned by now,” said European Commission President Ursula von der Leyen.

The E.U. has so far administered at least one dose to 26 percent of its total population, compared with 45 percent in the United States, 52 percent in Britain and 60 percent in Israel.

The E.U. is about seven weeks behind the United States on doses, adjusted for population size.

But even as the E.U. campaign takes off, the costs of the slow start are being felt. A bruising third wave struck the continent toward the end of winter, leading to scores of preventable deaths among the uninoculated elderly. Lockdowns caused the E.U.’s economy to contract in the first three months of the year. All the while, the bloc’s leaders were warring with AstraZeneca, the drugmaker that was falling short of delivery pledges, whose vaccine had initially been counted on to play a primary role in the campaign.

“Obviously, it was painful for E.U. leaders in January and February,” said Jacob Kirkegaard, a senior fellow at the German Marshall Fund of the United States. “There was an acute shortage, while the countries they like to compare themselves with didn’t have that problem. But now, it has changed. The scale-up of production in the E.U. has been extremely rapid.”

Pfizer has said it will be delivering 250 million doses in the second quarter of the year, a fourfold increase from the first three months. Italy, for instance, received 6.9 million doses of Pfizer in April, compared with 8.7 million doses during the previous three months combined.

The E.U. expects supplies to be boosted further as additional vaccines start to come online. After a delay to assess concerns about rare blood clots, the rollout of Johnson & Johnson’s one-shot vaccine is getting underway in Europe.

Another vaccine, made by the German company CureVac, could soon be approved, depending on the imminent results of a late-stage clinical trial. The CureVac vaccine is similar to those produced by Pfizer and Moderna, in that it uses mRNA technology.

The E.U. has scrambled to make up for lost time after an approach characterized by more caution and less urgency than that of the United States.

While the United States’ Operation Warp Speed spared little expense to help companies build factories and expand their production capacity at an early stage last year, the E.U. behaved more like an ordinary customer in early vaccine negotiations, haggling with drug companies over the price of doses and leaving it up to the drugmakers to figure out how to produce the promised supplies.

The E.U. team that negotiated the deals on behalf of the 27-nation bloc had no experience with the issue, since health issues have previously been the responsibility of individual countries. And it was plagued by a mix of demands from rich countries, such as Germany, that wanted to go big with the more expensive, newer-technology Pfizer and Moderna vaccines, and poorer ones such as Bulgaria, for which cost was a major issue and which preferred AstraZeneca’s cheaper and easier-to-store doses.

European policymakers also did not fully realize the extent to which barriers to vaccine exports out of the United States, Britain and India would put pressure on E.U. production to supply the world, leaving fewer doses to supply the Europeans themselves.

The result was that as Britain, the United States and a handful of other countries raced ahead with vaccinations starting in December, the E.U. badly lagged behind. At the last minute, AstraZeneca also said it would not be able to deliver nearly as many doses of its vaccine as it had promised, further slowing the E.U. effort.

Politicians in every country faced down furious citizens.

“France: Homeland of Permanent Delay,” read one headline in France’s Le Point magazine in early March, atop an angry opinion piece about the molasses-slow vaccination campaign there.

But behind the scenes, a shift was underway. Von der Leyen, a physician, had developed a productive relationship with Pfizer chief executive Albert Bourla, who seemed more capable than AstraZeneca’s leaders of expanding production capacity and meeting new and faster delivery targets, officials said.

And Thierry Breton, the top European Commission official in charge of Europe’s internal market and a former French finance minister and industrialist, started a whistle-stop tour of the continent’s 53 vaccine factories, speaking to workers on the production floor and to leaders of the factories about what they needed to sort out snags and speed doses.

When he was tasked in early February with overseeing the vaccine production effort, his team had little insight into what was happening with the complex process needed to make vaccines from start to finish.

“There was not really a good overview of the numbers, of where are the factories, of what is the capacity of the factories,” said one European official familiar with the situation, speaking on the condition of anonymity to discuss the closed-door process. Breton mapped out the process, and realized that there was a good chance that Europe already had the capacity to meet the 70 percent target in July, the official said.

Companies have gone to him with issues as basic as a shortage of plastic bags used in bioreactors, devices that are crucial for making the substances needed for the vaccines. He found another supplier and fixed the bottleneck, the official said.

Countries have also fine-tuned their strategies, making doses easier to come by. Germany, for instance, has allowed family doctors to administer vaccines, meaning people are no longer obliged to go to government-run vaccine centers. Italy, meantime, under new Prime Minister Mario Draghi, appointed a military logistics specialist, General Francesco Figliuolo, to overhaul the rollout.

Figliuolo has helped create some 1,000 new vaccination centers, nearly doubling Italy’s total. He has worked with the country’s most populous region, Lombardy, on a new vaccine reservations portal, after the region’s initial system kept crashing, leading in a few instances to empty vaccination centers. The general, in March, set a goal for Italy to administer 500,000 doses per day — something that seemed like a long shot at the time, when the nation was administering some 200,000 daily jabs.

Two days over the past week, Italy exceeded 500,000 doses. It has averaged 442,000 daily doses in the latest week.

Figliuolo, in a telephone interview, described his role as, “basically, fine-tuning the vaccination machine.”

“By June, we are counting on having such an inflow that we’ll further increase the speed of the machine,” he said.

Stefano Pitrelli, in Rome, contributed to this report.