One woman tweeted that the officer Tasered the unarmed woman twice and shot at her five times while she was on the ground.
The officer, identified as an 11-year police veteran, had been patrolling an apartment complex in Baytown and spotted Brown.
The officer knew the 45-year-old woman had outstanding warrants and started trying to arrest her, police said.
She is also heard screaming, "I'm pregnant". According to a neighbor of Turner's, she had complained about being harassed by this officer in the past. However, police say as the officer tased the woman, she then gained control of his taser and tried to fire it at the officer.
It has not been confirmed whether she was pregnant and a post-mortem will be carried out to establish if she was.
Police haven't released the woman's name or specified what warrants had been issued for her arrest.
The officer then shot at her several times, hitting her at least once in the face, police said.
In the video, the woman is heard saying: "You're actually harassing me" and "I'm actually walking to my house".
Per department policy, the officer has been placed on three days of paid administrative leave, the Baytown reported.
Witnesses said police administered CPR, but the woman died at the scene.
"It's a tragic event for everybody involved", Dorris said during an interview with local CBS affiliate KHOU.
Investigators are trying to contact whoever recorded the video, because the person was a witness to the shooting, the lieutenant said. "Our hearts go out to the family of the deceased".
The Harris County District Attorney's Office is investigating, and the department will conduct an internal review.
Floyd Rubin, who said he has two children with Turner, told CNN that authorities have not shared information about the shooting with him and his family. "It's extremely disrespectful for everybody involved, but that's the day and age we live in with social media".
Raquelle Cuellar, a resident in the apartment complex, told ABC 13 that the woman was often spotted walking her dog in the area.
The video of the police shooting can be seen on the next page.