Prada’s first 10 years as a public company have been patchy. But the family-controlled luxury brand has a sharper new look and leader-in-waiting.

On Thursday, Prada set profitability, revenue and e-commerce targets at its first investor day since 2014. The fashion label wants to grow sales to €4.5 billion, or $5.1 billion, within a few years, up from the €3.2 billion it made in pre-Covid 2019. Prada is aiming for a 20% operating margin, compared with 11% currently, and wants to roughly double the size of its e-commerce business...

Prada’s first 10 years as a public company have been patchy. But the family-controlled luxury brand has a sharper new look and leader-in-waiting.

On Thursday, Prada set profitability, revenue and e-commerce targets at its first investor day since 2014. The fashion label wants to grow sales to €4.5 billion, or $5.1 billion, within a few years, up from the €3.2 billion it made in pre-Covid 2019. Prada is aiming for a 20% operating margin, compared with 11% currently, and wants to roughly double the size of its e-commerce business as a percentage of overall sales.

Patrizio Bertelli, Prada’s chief executive and husband of the brand’s designer Miuccia Prada, discussed retiring over the next three years and handing the reins over to his son, Lorenzo Bertelli. The younger Mr. Bertelli, who joined the board earlier this year after taking over the brand’s digital communication strategy in 2017, gave presentations on Prada’s e-commerce and sustainability plans at the event.

Tight family control hasn’t worked in investors’ favor so far at Prada. Since making its market debut in Hong Kong in 2011, the stock has delivered annual total shareholder returns of 6%—well below the roughly 20% annual gains of LVMH and Hermès, which are also dominated by their family founders.

Mr. Bertelli and Mrs. Prada underestimated the importance of e-commerce, which contributed just 2% of sales in 2019, although this number has since jumped to 7%, so far this year. With a small free float, minority investors in the company have had little power to push for a shake-up in the management or design teams. However, changes are happening. Prada is investing in online technology, hired a new co-chairman six months ago and brought in co-designer Raf Simons.

The goals set on Thursday aren’t especially ambitious—Prada used to make an operating margin of 27% in 2012, for example—but there is a clear plan to achieve them. Selling more goods directly to consumers rather than through department stores and other third parties should boost profitability. Prada is taking costs out of its supply chain and managing its inventory more efficiently.

The brand has an opportunity to grow in the booming U.S. luxury market, even if it is late to the party. Prada makes around 14% of overall sales in the Americas, compared with 22% for Cartier’s owner, Richemont. The Americas are now the world’s biggest luxury market by value, making up 31% of the global luxury sales compared with 21% for China, according to consultancy Bain & Co.

Reducing the brand’s dependence on China seems wise, even if there is no sign of a slowdown in Chinese luxury spending. Between mid-August and late-September, Prada’s shares lost 28% of their value after China’s president, Xi Jinping, gave a speech hinting at greater wealth redistribution in the country. They have since recovered, but investors would probably welcome a more balanced look.

Considering how well known the brand is, Prada is still a surprisingly small business. Even if the label hits its new sales target, Prada will be less than half the size of Christian Dior,

which is just one of 70-plus luxury brands in LVMH’s portfolio. Investors pay a ritzy 48 times projected earnings to own the Italian label’s stock. Prada finally looks ready to measure up.

Write to Carol Ryan at carol.ryan@wsj.com