Turner was mayor of Houston during the weekend of May 29-31, 2020, when events and activity took place in cities across the U.S. following the death of George Floyd. Events in Houston, Texas, began on Friday, May 29, 2020, with a march downtown to city hall. No curfews were issued. The national guard was not deployed. On May 25, 2020, Minneapolis police officers arrested Floyd, a Black man, after receiving a call that he had made a purchase with a counterfeit $20 bill. Floyd died after Derek Chauvin, a white officer, arrived at the scene and pressed his knee onto Floyd's neck as Floyd laid face-down on the street in handcuffs. Both the Hennepin County Medical Examiner and an independent autopsy conducted by Floyd's family ruled Floyd's death as a homicide stemming from the incident. The medical examiner's report, prepared by Dr. Michael Baden and Dr. Allecia Wilson, said that it was "not a legal determination of culpability or intent, and should not be used to usurp the judicial process." On April 20, 2021, Chauvin was found guilty of murder and manslaughter in relation to Floyd's death. Floyd's death was filmed and shared widely, leading to activity regarding racism, civil rights, and police use of force. The first events took place in Minneapolis-St. Paul on May 26. An event in Chicago organized by Chance the Rapper and Rev. Michael Pfleger took place the same day, making it the first major city outside of Minneapolis to host an event in response to Floyd's death.

Turner was mayor of Houston during the weekend of May 29-31, 2020, when demonstrations and protests took place in cities across the U.S. following the death of George Floyd. Protests in Houston, Texas, began on Friday, May 29, 2020, with a march downtown to city hall. No curfews were issued. The national guard was not deployed. On May 25, 2020, Minneapolis police officers arrested Floyd, a black man, after receiving a call that he had made a purchase with a counterfeit $20 bill. Floyd died after Derek Chauvin, a white officer, arrived at the scene and pressed his knee onto Floyd's neck as Floyd laid face-down on the street in handcuffs. Both the Hennepin County Medical Examiner and an independent autopsy conducted by Floyd's family ruled Floyd's death as a homicide stemming from the incident. The medical examiner's report, prepared by Dr. Michael Baden and Dr. Allecia Wilson, said that it was "not a legal determination of culpability or intent, and should not be used to usurp the judicial process." Floyd's death was filmed and shared widely, leading to protests and demonstrations over racism, civil rights, and police use of force. The first protests took place in Minneapolis-St. Paul on May 26. A protest in Chicago organized by Chance the Rapper and Rev. Michael Pfleger took place the same day, making it the first major city outside of Minneapolis to host a protest over Floyd's death.

On May 14, 2020, Turner announced he tested negative for COVID-19. He got tested after fellow council member Letitia Plummer tested positive for the virus.