On Wednesday, Turner spoke at the city's newest mobile testing site at Worthing High School in the Sunnyside neighborhood, which was mentioned as a potential hotspot for the spread of the virus.
The site requires appointments since there are only 150 tests available, Turner said.
Immediately after Gov. Greg Abbott laid out a plan to reopen certain businesses with a 25 percent capacity limit, Turner said on Monday that he will respect the superseding order from Austin, but with emotion, the mayor stated his job is still about protecting particularly disadvantage residents.
At Worthing, Turner announced two new virus deaths, bringing the city's toll to 52. All but one of the deaths involved a person with underlying medical issues.
Earlier in the week, the city already opened a new testing site in southeast Houston for anyone regardless of symptoms.
The day before Harris County Judge Lina Hidalgo said county test sites will expand to up to 1,600 tests.
All of this currently weighs on leadership's minds before Thursday's "Stay Home, Work Safe" order expires.
Just before Turner spoke at Worthing, the mayor and city council grappled over a potential trash bin fee that would help with the impending budget shortfall in the midst COVID-19.
The item will be further discussed next week, but during the morning's city council meeting, a testy exchange was captured between the mayor and one of his colleagues.
WATCH: Fiery exchange during Houston city council meeting over trash bin fee