BEIJING—Several small, privately held banks in China, heralded at their launch two years ago as challengers to state lending giants, are struggling to make inroads among borrowers.

Data from China’s banking regulators released last week show that old-line banks dwarf the five newcomers, which include WeBank and MYbank, the online banks set up, respectively, by technology majors Tencent Holdings Ltd. and Alibaba Group Holding Ltd.’s affiliate Ant Financial Services Group.

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