Houston officials have accused Uber of aggressive behaviour as the car service start-up and its rivals attempt to enter the city and shake up the taxi market.

Smartphone app-based companies such as Uber and Lyft have expanded into dozens of major cities in recent years, promising more efficient service than traditional taxi and limousine firms but prompting local politicians to question whether the rides are legal.

Uber is available in 32 countries and close to 40 North American cities but has been fiercely opposed by established businesses and banned in places including Portland, New Orleans and Miami - much to the frustration of one of its investors, Ashton Kutcher.

Houston, the US’s fourth-largest city, is the biggest hold-out remaining in North America, and its size and limited public transport make it a prime target. The Texas city’s council members are expected to spend up to several months considering whether to alter regulations to permit the technology businesses to operate, but officials have been unhappy with the tactics of the two best-known brands.

Uber and Lyft have started to offer rides in Houston for free on a promotional basis, arguing that as long as no money changes hands they are not violating city rules. “Uber and Lyft decided to up the pressure by entering the market and operating legally because they’re not accepting fares. But they’re trying to bully us into acting,” Houston mayor Annise Parker told the Guardian.

Parker, a former gay rights activist, said she would not be browbeaten. “Uber cannot threaten me in any way. I had people threatening to shoot me in the seventies, there’s not anything that Uber as an entity can do that will intimidate me,” she said.

“It just slows down the process because what happens - I know there are politicians who will cave under pressure - but often what happens for the majority of us when someone starts applying that kind of pressure, you don’t want to look like you cave in to pressure so you slow down and get dug in. And it’s not I that have to make this decision alone, there are 16 [council members]. It has not made it easier.”

Uber claims that more than 10,000 people have signed a petition supporting it. Houston’s city attorney, David Feldman, sent an email to Uber last month formally asking that it “cease and desist” from encouraging the public to write to officials demanding the introduction of the service. “The excessive number of e-mails has gone unabated, to the point that it has become harassing in nature and arguably unlawful,” Feldman wrote.

“It’s not that we mind hearing from our citizens, but these internet petitions ... while I respect that someone who lives in New York City really really likes Uber, they’re not my constituents,” Parker said. “And having 300 emails hit my inbox from people who don’t live in Houston and then I can’t send or receive emails from someone who may really need help is a distraction. But it has abated somewhat.”

Nairi Hourdajian, a spokesperson for Uber, said: “We were giving voice to all of the people in Houston, more than 10,000, who have been clamouring for more and better transportation options... we think it would be a particularly strong market.” She added that the company’s research indicated it would be popular with many drivers. Lyft did not respond to a request for comment.

Kitessa Turi, of the United Houstonian Taxi Drivers Association, said that many cab drivers in the city struggle to make a living. He is unsure if the start-ups would help or hurt his group’s members. “I’m not against anybody, I support the open market, but we have no idea how [they] want to conduct business,” he said.

Founded in 2009, Uber is backed by investors including Goldman Sachs and Google Ventures. It offers a range of services. UberX is a ride-sharing system, UberBlack provides sedans and UberLux is the company’s high-end offering. The app allows users to order a car and watch on a map as it heads to the pick-up point. Often more expensive than a standard taxi, “surge pricing” means that customers pay more at times of peak demand, such as in bad weather.

Also headquartered in San Francisco, Lyft is a ride-sharing scheme that claims to be generally cheaper than taxis. In some cities it charges set amounts for journeys and in others its drivers accept “donations”.

New digital companies such as Uber and the accommodations website Airbnb are at the forefront of the “sharing economy”, which challenges traditional business models by encouraging peer-to-peer marketplaces.

However, innovation and regulation have not always been easily compatible. Insurance coverage is a problematic issue for the car start-ups. Regulators in US cities have examined whether ride-sharing drivers should each require commercial insurance policies.

Uber was sued in January by the family of a six-year-old girl who was killed in San Francisco by a driver linked to the company. “I am a little concerned at the way, the cavalier attitude, that - UberBLACK would be the example I use - they contract directly with drivers of black town cars. Most of those drivers, virtually all of those drivers, do not actually own the vehicles that they drive, at least here in Houston,” said Parker, they mayor.

“And so here’s a real-world problem. Uber contracts with the driver, they give the driver their smartphone, the driver starts moonlighting in-between regular limo runs for their company, that driver has an accident. Whose insurance covers that?... It’s not as easy as just saying: ‘we’ll let them come in and let the market sort it out’. I’m not worried about the limo company, not necessarily worried about the driver, I’m worried about the customer in that car and whether they’re safe.”

Uber faces a class-action lawsuit from drivers related to its tipping charge. Resistance to its arrival in Washington was so virulent that a filmmaker made a documentary called The Uber Wars. And it upset Salman Rushdie. But Uber’s financial muscle is allowing it to wage extensive PR campaigns and hire influential lobbyists. It marketed aggressively and organised a petition with more than 10,000 signatures in Seattle, where city council members voted last month to limit the number of drivers. Despite its divisive tactics and controversial business model, Uber usually gets its way when it tries to expand and it seems likely that Houston will be no different in the end. “We have successfully deregulated all sorts of industries, and I’m assuming that we will figure out a way to work Uber in down here,” said Parker.