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Bailey said the measures would be "imposed" regardless of the outcome of a consultation that is currently under way into social media's harms for children.
The concession by the government comes after pressure from the House of Lords over the issue, led by Tory former education minister and academy chain founder John Nash.
MPs voted four times to press the Commons into accepting a ban.
The government had already begun a consultation on access to technology for under-16s, and had floated measures such as age restrictions on social media as well as other services such as gaming sites and artificial intelligence chatbots.
Nash described the government's concession as "a huge step forward for our children's safety online".