The fight and shooting happened in the 1600 block of Garth Road at about 10:40 p.m., according to Lt. Steve Dorris with the Baytown Police Department.
Dorris told reporters Monday night that the woman was struck by at least one bullet and died at the scene.
A police officer, who has not been identified, tried to arrest a woman who had outstanding warrants.
Cell phone video shows Turner yelling at the officer: "I'm walking!"
A "pop" can be heard - which came from a police stun gun, according to Good Morning America - and a struggle ensues.
It has not been confirmed whether she was pregnant and a post-mortem will be carried out to establish if she was.
"The female was able to get the officer's Taser away from him", he said, according to KPRC.
"You're actually harassing me", Turner says after she managed to break free from the officer's grip. The woman is on the ground and appears to reach toward the officer when she seems to be heard saying: "I'm pregnant".
She yells, "I'm pregnant!"
He added, "It's a tragic event for everybody involved".
The unidentified officer was patrolling the area when he saw the woman-identified by family as Pamela Turner-who he apparently recognized from "prior dealings".
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The officer involved is an 11-year veteran of the Baytown Police Department.
The Harris County District Attorney's Office is investigating, and the department will conduct an internal review.
The officer is now on administrative leave, in accordance with standard protocol.
Dorris has said that in addition to this bystander video, there is likely also video from the officer's bodycam.
The shooting is under investigation. "It was a sad situation", Cuellar said.
"It's unfortunate that someone takes a tragic incident like this and starts posting it on social media", Dorris told KHOU.
Residents of the apartment complex told the station that Turner was harmless and that she typically just walked around the neighborhood with her pet. "It's extremely disrespectful for everybody involved".
The video of the police shooting can be seen on the next page.
Moments later he killed her.
Police are searching for the person who recorded the video, with Dorris denouncing the action.