In her own words, the car was going fast. From this CNN article

The car kept on the road, going under an underpass full of puddles and almost losing control to avoid them. We all incredibly laughed. It was liberating. Losing control of the car in a street full of water in Baghdad and maybe wind up in a bad car accident after all I had been through would really be a tale I would not be able to tell. Nicola Calipari sat next to me.

In this BBC interview, she said she was fired on by tanks.

We were on our way to the airport when the tanks started to strike against us and he tried to cover me and he was shot. He died and, me, I was safe but he was dead.

In this Guardian article, she claims that hundreds of rounds were fired at her car.

Italian reconstruction of the incident is significantly different. Sgrena told colleagues the vehicle was not travelling fast and had already passed several checkpoints on its way to the airport. The Americans shone a flashlight at the car and then fired between 300 and 400 bullets at if from an armoured vehicle.

If hundreds of rounds were fired at her car, it would've looked like what happened to Bonnie and Clyde. No one in that car would likely have survived, much less 3 out of the 4 onboard.

Now, it seems she's saying she was fired upon by helicopters. Seeing as how Apache helicopters carry a 30mm antitank cannon, Cobra helicopters carry a 20mm cannons, and other helicopters typically carry 7.62mm miniguns capable of firing 6000 rounds per minute, the chances of anyone surviving a helicopter attack against a car seem unlikely.

Likewise, if it was a tank, you'd be talking about receiving fire from at best a 7.62mm machine gun and quite likely a .50 caliber machine gun, then the chances of survival aren't too great if they were trying to kill people inside a car. Bradley APCs carry 25mm cannons with enough power to take out armor.

If this story is accurate, the Italian press has reported that their agents apparently kept the CIA in the dark about the release.

Italian agents likely withheld information from U.S. counterparts about a cash-for-freedom deal with gunmen holding an Italian hostage for fear that Americans might block the trade, Italian news reports said yesterday.

The decision by operatives of Italy's SISMI military intelligence service to keep the CIA in the dark about the deal for the release of reporter Giuliana Sgrena, might have "short-circuited" communications with U.S. forces controlling the road from Baghdad to the city's airport, the newspaper La Stampa said.

That account may be confirmed from this otherwise hostile Washington Post article

The automobile was traversing onto a route -- the road to the airport -- where soldiers have been killed in shootings and by roadside bombs. U.S. soldiers had established an impromptu evening checkpoint at the entrance to the road about 90 minutes earlier and had stopped other vehicles. They knew a high-level embassy official would be moving to the airport on that road, and their aim was to support this movement.
But no specific coordination occurred between those involved in Sgrena's rescue and the military unit responsible for the checkpoint, according to the source, who said he cannot be named because the military's investigation into the incident is continuing.

Soldiers at the checkpoint have told U.S. military officers that they flashed lights, used hand signals and fired warning shots in an effort to stop the car, which they believed was traveling at more than 50 mph, a typical speed for that road. But Sgrena, who had just been released by Iraqi captors, recalled later that the car was not traveling very fast and that soldiers started firing "right after lighting" a spotlight -- a decision she said was not justified. Sgrena was wounded by shrapnel in the U.S. barrage.

The absence of advance communication between the Italians and the U.S. soldiers at the checkpoint appears to have put the occupants of the car in grave jeopardy, given what many U.S. officials describe as the military's standard practice of firing at onrushing cars from their checkpoints in Iraq.

"In my view, the main contributing factor was a lack of prior coordination with the ground unit," the source said. "If requested, we would have resourced and supported this mission very differently."

A couple years ago, Mary Anne made a reasonable assertion about cops. She said something to the effect that, "If the cops have to chase you down, you're quite likely going to get a beating."

Likewise, if you're traveling on one of the most dangerous highways on Earth and don't stop at a checkpoint, you're going to get fired upon. It was tragic, but so were the deaths of many military members killed by suicide bombers at similar checkpoints.