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.

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.

“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.”