Turner officially relinquished her US citizenship on October 24th when she signed the “Statement of Voluntary Relinquishment of U.S. Citizenship under Section 349 (a)(1) of the INA” — the Immigration and Naturalization Act.

Relinquishment is different than renouncing citizenship.

Washington Post:

The key word in the embassy report apparently is the term "relinquishment." That means, a knowledgeable source told us,  that she did not "formally renounce her U.S. citizenship under 349(a)(5) Immigration and Nationality Act,  but took Swiss citizenship with the intent to lose her U.S. citizenship."

As opposed to formal renunciation — a much more complex process, we were told — there are  no "tax or other penalties for loss of citizenship in this fashion."

Turner  has lived in Switzerland for nearly two decades. In July, she married her  boyfriend of 27 years,  German music producer Erwin Bach (unclear if related to Johann Sebastian).  Turner had taken the oath of Swiss nationality April 10.  She's fluent in German, the report said, and she declared that she no longer has any strong ties to the United States "except for family, and has no plans to reside in the United States in the future."