The oft-heard claim that charters perform no better than conventional schools is out of date and inaccurate.

Many have been puzzled by New York Mayor Bill de Blasio's skepticism toward charter schools, his calls for ending space-sharing and charging them rent, and his $210 million cut of a construction fund important to the schools. Education reformers are also anxious about the failure of President Obama and Education Secretary Arne Duncan to defend charter schools in the face of these prominent reversals of New York City policy. Is this just about teacher-union politics, or are there perhaps legitimate performance reasons for tapping the brakes on charter schools in public education today?

The first thing to remember about charter schools is how recent an invention they are. Born in the 1990s, it wasn't until 2006 that total enrollment reached a million children—out of 55 million pupils in the country. More than half of the charters in New York City are less than five years old.

With huge waiting lists for every available seat, though, charters are now beginning to mushroom. Well before Mr. de Blasio faces re-election in 2017, charters will educate 10% of New York City's public-school students, and they already enroll a quarter of all pupils in some of the city's poorest districts. Nationwide, charter schools will enroll five million by the end of this decade.

But do they get results? Initial assessments were mixed. In the early days, charter authorizing was very loose, nobody knew what worked best, and lots of weak schools were launched. The system has since tightened. In Washington, D.C., for instance, seven out of nine requests to open new charters are now turned down, and 41 charters have been closed for failing to produce good results.

Nationwide, 561 new charter schools opened last year, while 206 laggards were closed. Unlike conventional public schools, the charter system allows poorly performing schools to be squeezed out.

As charter operators have figured out how to succeed with children, they are doubling down on the best models. Successful charter schools have many distinctive features: longer school days and longer years, more flexibility and accountability for teachers and principals, higher expectations for students, more discipline and structure, more curricular innovation, more rigorous testing. Most charter growth today is coming from replication of the best schools. The rate of enrollment increase at high-performing networks is now 10 times what it is at single-campus "mom and pop" academies.

The combination of weak charters closing and strong charters replicating is having powerful effects. The first major assessment of charter schools by Stanford's Center for Research on Educational Outcomes found their results to be extremely variable, and overall no better than conventional schools as of 2009. Its follow-up study several years later found that steady closures and their replacement by proven models had pushed charters ahead of conventional schools. In New York City, the average charter-school student now absorbs five months of extra learning a year in math, and one extra month in reading, compared with counterparts in conventional schools.

Other reviews show similar results, and performance advantages will accelerate in the near future. Charter schools tend to start small and then add one additional grade each year. Thus many charters in New York and elsewhere are just getting started with many children. As the schools mature, and weak performers continue to be replaced, charters will become even more effective.

But the results top charter schools are achieving are already striking. At KIPP, the largest chain of charters, 86% of all students are low-income, and 95% are African-American or Latino, yet 83% go to college. In New York City, one of the academies Mr. de Blasio has denied additional space to is Harlem's highest-performing middle school, with its 97% minority fifth-graders ranking No. 1 in the state in math achievement. It and the 21 other schools in its charter network have passing rates on state math and reading tests more than twice the citywide average.

Judged by how far they move students from where they start, New York charter schools like Success Academies, Uncommon Schools, Democracy Prep and Achievement First—and others like them across the country—are now the highest-achieving schools in America. The oft-heard claim that charters perform no better than conventional schools on the whole is out of date and inaccurate.

Remarkably, charters do all this on the cheap. In a city where conventional public schools spend $19,770 per student, the New York City Department of Education funded its public charter schools at only $13,527 per pupil in the latest year. That's right around the average disparity nationwide, where urban charter schools get 72% of what conventional public schools receive for each child enrolled.

When the next school year starts this fall, there will be nearly 7,000 charter schools in America, with the growth curve pointing sharply upward. Historians who look back at our era may describe charter schools as the most consequential social invention of this generation, with potent effects on economic mobility.

And chartering represents one of the great self-organizing movements of our age. It rose up in the face of strong resistance from the educational establishment. It has been powered by independent social entrepreneurs and local philanthropists. It is a response by men and women who refused to accept heartbreaking educational failures that the responsible government institutions showed no capacity to solve on their own.

Mr. Zinsmeister is the author of "From Promising to Proven: A Wise Giver's Guide to Expanding on the Success of Charter Schools," just published by The Philanthropy Roundtable.

This article was published in the Wall Street Journal Online on March 28, 2014.