Large stores stacked with thousands of choices can become “paralyzing” for consumers, said Andre Javes, chief executive of the Asia business. “That was always the challenge in some of the Western countries.”

The toy store was doomed in the U.S. by a hefty debt load from a 2005 leveraged buyout, the rise of e-commerce and its model of building big-box outlets in out-of-town locales, which struggled to keep pulling in customers. Impatient creditors pushed it over the edge.

After filing for bankruptcy protection in the U.S. in September 2017, Toys ‘R’ Us Inc. was unable to restructure its debt and save its U.S. operations. The company liquidated all its U.S. stores and sold off its Canadian and European units. Its Asian operations were auctioned off and acquired earlier this month by its Hong Kong-based partner, Fung Retailing Ltd., and a group of investment firms.

The newly independent Toys ‘R’ Us Asia said it would stick to smaller shops of around 10,000 square feet compared with the 40,000 square-foot average in the U.S. The stores are often located inside malls and target buyers looking for high-quality, educational toys.

Although revenue in Toys ‘R’ Us’s combined businesses in Japan, China and Southeast Asia fell for four straight years through 2016, it rebounded to $1.6 billion in the 12 months to Jan. 28 2017, the last data available from Toys R Us Inc.’s public filings. That put sales ahead of Europe and second only to the U.S.

The surviving Asian operation spans more than 450 stores in 10 markets, with 168 in Japan and 166 China. It intends to open 68 new stores in 2019, with around 50 of them in mainland China, Mr. Javes said.

China’s market is particularly lucrative. Spending on children’s products there is projected to rise to $189 billion in 2022, from $142 billion in 2017, according to research firm Mintel. Higher incomes will increase spending power for parents to buy more expensive products for their children, Mintel said.

Toys ‘R’ Us and other brick-and-mortar stores still face a fight to grow as e-commerce giants Alibaba Group Holding Ltd. and JD.com have taken a huge swath of China’s retail market. More people bought toys, games and books online than from specialist children’s retail stores in the second half of last year, according to a study of 3,000 internet users by Mintel.

Toys ‘R’ Us Asia is also pushing online sales, but doesn’t believe e-commerce will capture all of consumers’ spending. “You need to have a hybrid,” said Mr. Javes.

Across Asia, over 40,000 new customers a week join Toys ‘R’ Us’s customer loyalty program, which now numbers roughly 20 million members, Chief Commercial Officer Jo Hall said. Glossy pamphlets advertise a top 20 list of company mascot Geoffrey’s suggested toys—like Harry Potter-themed merchandise and classic board games such as Hasbro Inc.’s Monopoly.

A lack of commercial retail space in Asia’s dense urban centers has also led Toys ‘R’ Us Asia to display a more carefully curated range of toys because stores are smaller. One focus has been on development-related toys, as parents become more focused on their children’s education.

The company is also seeking to lure in shoppers through in-store activities. At a Toys ‘R’ Us store on the seventh floor of a Hong Kong mall in a busy shopping district, several 5-year-olds played on anime-themed arcade machines, with parents gathered behind.

“My daughter is always asking to come and play. It’s almost like a park for her,” mother Angel Wong said.

Another factor helping Toys ‘R’ Us is that some consumers remain wary of buying toys online, especially in China, which has long been plagued by counterfeits, as well as safety scandals around children’s products.

“There’s better guarantee for products’ quality,” said Sam Yeung, a mother of two, as she and her 4-year-old son checked out interactive globes and action figures. “For many products sold online, you never know if it’s real,” she said.

—Yifan Wang contributed to this article.

Write to Ese Erheriene at ese.erheriene@wsj.com