Paris is cracking down on electric scooters after an anarchic year in which startups flooded the city’s streets with their vehicles.

The French capital has 12 free-floating scooter operators, more than the entire US, according to a study released on Thursday, and about 20,000 of the two-wheeled vehicles.

Broken scooters end up in some of the city’s famed gardens or are even tossed over bridges into the Seine, and the city’s pavements have become something of a battleground between riders and pedestrians.

Mayor Anne Hidalgo says electric scooters have fallen into a legal grey zone and after repeated complaints and a spate of injuries and near misses, the mayor and police want to limit speeds to 20km/h (12mph) in most areas and 8km/h (5mph) in areas with heavy foot traffic, and prohibit parking anywhere but designated spaces. Hidalgo also plans to limit the number of operators to three and cap the number of scooters.

The city already imposes €135 (£120) fines for riding on the pavement and €35 (£31) fines for blocking the pavement while parked. Operators are charged by the city for any broken scooters that need to be picked up by municipal workers.

“We need order and rules to assure road safety and to calm the streets, pavements and neighbourhoods of our city,” Hidalgo said on Thursday.

Isabelle Vanbrabant was knocked over by an electric scooter in April, and suffered multiple fractures to her right arm. Photograph: Nadine Achoui-Lesage/AP

For Isabelle Vanbrabant, any regulations are too late. The pianist at Paris’s famed opera was coming home from work last month and walking across a square near Les Halles when a rider on an electric scooter came up from behind, knocking her over and continuing on his way.

She fell on her right arm, suffering multiple fractures. She yelled for the rider to return, which he then did, and to call for help. However, her prognosis is uncertain.

“I need this hand, I need to have it back,” she said.

About a third of electric scooter riders in France are foreign tourists, according to the study by 6t-bureau, a research consultancy specialising in urban planning and transport.