While that's certainly one of the reasons child labour was drastically reduced in the U.S., it wasn't the only one. I think history.com's article on child labour is good:
https://www.history.com/topics/industrial-revolution/child-labor
Note that I said drastically reduced, not eliminated. Some sobering information from the article:
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Does Child Labor Exist Today?
Although child labor has been significantly stalled in the United States, it lingers in certain areas of the economy like agriculture, where migrant workers are more difficult to regulate. Since 1938, federal laws have excluded child farm workers from labor protections provided to other working children. For example, children 12 and younger can legally work in farm fields, despite the risks posed by exposure to pesticides and farm machinery.
Employers in the garment industry have turned to the children of illegal immigrants in an effort to compete with imports from low-wage nations. Despite laws limiting the number of hours of work for children and teens still attending school, the increasing cost of education means many are working longer hours to make ends meet. State-by-state enforcement of child labor laws varies to this day.
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Also important to remember that we're only talking about the U.S. here. I think unicef makes a good point that not all child labour is bad (My parents paid me and my siblings to do dishes as a kid for pay and I still think it was a good thing to be taught early on that doing work can get you monetary rewards), but that some definitely is and is still quite common around the world:
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Children around the world are routinely engaged in paid and unpaid forms of work that are not harmful to them. However, they are classified as child labourers when they are either too young to work or are involved in hazardous activities that may compromise their physical, mental, social or educational development. In the least developed countries, slightly more than one in four children (ages 5 to 17) are engaged in labour that is considered detrimental to their health and development.
The issue of child labour is guided by three main international conventions: the International Labour Organization (ILO) Convention No. 138 concerning minimum age for admission to employment and Recommendation No. 146 (1973); ILO Convention No. 182 concerning the prohibition and immediate action for the elimination of the worst forms of child labour and Recommendation No. 190 (1999); and the United Nations Convention on the Rights of the Child. These conventions frame the concept of child labour and form the basis for child labour legislation enacted by countries that are signatories.
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Source:
https://data.unicef.org/topic/child-protection/child-labour/
