"We are taking this very, very seriously ... but take a collective breath," Turner said. "I don't want the collective fear to do more harm than the virus itself."
The city has been coordinating with major hospital leaders to ramp up testing, he said.
"The Texas Medical Center has the capacity to take care of those coming to their respective hospitals," Turner said. "We're talking about going beyond the hospital walls, testing in the community."
He said it would involve multiple sites throughout the city and would be accessible to as many people as possible, based on need. People who think they should be tested should contact their doctor or the city's coronavirus hotline, 832-393-4220.
The update came as the city confirmed it has its fourth presumptive positive case of coronavirus disease, or COVID-19. Officials acknowledged that the number of cases will rise with additional testing, while other active cases could be removed as patients recover.
"If they come in small numbers every day, that’s what we want to happen," EMS Director David Persse said.
Despite widespread event cancellations and facility closures, Turner said the city remains open for business and is not advising restaurants and shops to close.
"What we're saying is, be responsible," he said.
The mayor advised leaders of churches, synagogues, mosques and other congregations to use their own judgment in deciding whether to hold religious services after the decision to bar events with more than 250 people.
"We're not giving any directive," he said, but he did make these requests: no handshaking, no embracing and exercise social distancing.