He will be eligible to return October 1, ahead of the team's Week 5 game at New England on Oct. 4.
"A urine sample I provided to the NFL has resulted in a positive test of a banned substance under league policy with regards to Performance Enhancing Substances. It absolutely kills me that I allowed this to happen," Turbin wrote on Twitter. "It has always been a priority in my life to display a high level of leadership both on and off the field. I strive to be a person that people can look up to and strive to exemplify the definition of hard work.
"Unfortunately, I have made a mistake. A mistake that will not only have personal consequences, but will affect the Colts organization, an organization that I am extremely appreciative to be a part of. For that, I am truly sorry."
Turbin, who is entering his eighth season, spent the majority of the offseason program working with the first unit after the Colts decided not to re-sign veteran Frank Gore last winter and with Marlon Mack out with a shoulder injury.
Turbin played only six games last season because of an elbow injury.