According to the 1988 edition of the Morris Dictionary of Word and Phrase
Origins:
Normal Schools derive their name from the French phrase ecole normale. These
teacher-training institutions, the first of which was established in France
by the Brothers of the Christian Schools in 1685, were intended to set a
pattern, establish a “norm” after which all other schools would be
modeled. The first normal school in America was established in Vermont in
1823. The name fell out of favor toward the end of the 1920s, when the
influence of Columbia University’s Teachers College became paramount in the
field of public education. Most such institutions changed their names to “
teachers colleges” during the 1930s. Now that the “progressive education”
teachings of the Columbia group have been discredited, the Progressive
Education Association itself has disbanded and most colleges have dropped “
teachers” from their names. Thus we find that the normal school of
grandfather’s day became a “state teachers college” during father’s
youth, but today’s sprouts are attending “state colleges.”
https://www.jmu.edu/centennialcelebration/normalschool.shtml#:~:text=Normal%20Schools%20derive%20their%20name,other%20schools%20would%20be%20modeled.