The 23-year-old, who was released by the Magpies earlier this month, is the Gills' second keeper signing this week.
Charlton Athletic stopper Ashley Maynard-Brewer joined the club on a season-long loan on Tuesday.
Turner began his career in the academy at Bolton Wanderers before signing for Newcastle in the summer of 2019.
He joined Morecambe on a season-long loan in the summer of 2020, making 17 appearances and keeping three clean sheets before being recalled to Tyneside in early 2021.
A loan at Colchester followed later that year, where he made 14 appearances.
The length of Turner's contract has not been specified.