According to Merriam-Webster, this is the correct way to spell new-fangled.

That which displays newfangledness (also a real word, if you can believe it); meaning of the newest kind or style. A much more popular word than oldfangled. In fact, the etymology of oldfangled dates back to about 1842 which makes it a rather oldfangled term in its own right.

New"fan`gled (?), a.

1.

Newmade; formed with the affectation of novelty.

"A newfangled nomenclature."

Sir W. Hamilton.

2.

Disposed to change; inclined to novelties; given to new theories or fashions.

"Newfangled teachers." 1 Tim. vi. (heading). "Newfangled men." Latimer.

 

© Webster 1913.

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