"While I desire and require a full investigation, I am confident that once complete, the public will conclude we have been transparent," Turner said in a statement.
Houston Police Department officers shot and killed Braziel, 38, on Cullen Boulevard early Saturday morning. Officers said they observed Braziel standing in the middle of the street holding a handgun. Police said Braziel ignored commands to drop the weapon and, fearing for their safety, they shot him dead.
In his statement, Turner recommended police release all available surveillance footage of the shooting to the public, because "tensions are running high."
The mayor said he watched video footage from a nearby gas station and saw Braziel fire two shots into the air as police, in a cruiser, pulled up to the scene. In footage yet to be released publicly, Turner said a bystander acknowledged Braziel had a gun, which police recovered from his body.
Turner has repeatedly urged calm in the wake of three high-profile shootings across the country last week: the shooting deaths of black men by police in Minnesota and Louisiana, and an ambush of police by a sharpshooter in Dallas that killed five officers.
Minnesota governor Mark Dayton asked the Department of Justice to investigate the death of Philando Castille. DOJ officials announced an investigation into the shooting death of Alton Sterling by Baton Rouge, Louisiana, officers, after that state's governor, John Bel Edwards, promised transparency.
While state and local officials are eager to dispel allegations of biased policing, the federal government has limited jurisdiction to investigate and prosecute officer-involved shootings, should they be unjustified. As shooting-related crimes such as murder and manslaughter are prosecuted at the state level, federal investigators may only examine if police willfully violated a person's civil rights, often a much harder charge to prove.
The Department of Justice can, however, mandate reform of local police departments if investigators find systemic failures — as the feds did in Albuquerque, New Mexico, two years ago.