“I felt like I was going to die,” she said.

But when Ms. Karna got past the worst of her trauma, she could find out almost nothing about the crash. All she had was the New York Police Department’s two-page report, filled out by the officer who responded to the scene. On it, a box labeled “Not Investigated at Scene” is checked, which meant the department’s Collision Investigation Squad, a specialized unit that collects evidence and determines whether a crime occurred, had not come to the scene of the crash.

The Collision Investigation Squad was once an obscure division that was unknown outside the Police Department, and barely known within. Then, starting in 2011, two fatal crashes in quick succession brought public scrutiny and withering criticism to the unit for its incomplete and toothless investigations.

The Police Department vowed to do better. By 2013, the department had added eight officers to the crash squad, bringing the total to 26, and it established a new team of 12 crime scene technicians to assist the collision investigators. Commissioner Raymond W. Kelly said he believed the newly fortified unit would nearly quadruple its caseload, investigating as many as 1,200 crashes each year.

Cases like those of Ms. Karna were exactly the sort that the squad seemed to be created to handle.

This August, more than two years after she was nearly killed on Third Avenue, Ms. Karna appeared before City Council to share her story and testify in favor of two bills to study and deter dangerous driving habits. She also expressed dismay at the Police Department’s failure to hold anyone responsible for her crash.

“It’s unacceptable,” Speaker Corey Johnson said after Ms. Karna’s testimony. “That would be shocking and upsetting, allowing someone who … almost killed Bernadette to continue to get back on the streets of New York City without a thorough investigation.”