The correct word to use here depends on what, exactly, happened.
If the car lost control, then skid likely describes the situation best. There may be specific kinds of skidding that could also apply.
If the driver, for whatever reason, suddenly jerked the wheel and changed the car’s direction, then swerve is the verb for that.
If the driver wasn’t paying attention, and gradually moved out of their intended lane into somewhere they didn’t belong, then drift could be used.
Alternatively, if the driver was attempting to intentionally skid around a corner at high speed as a racing maneuver, that would also be drifting.
Skid refers to losing control due to insufficient traction:
(esp. of a vehicle) to slide unintentionally on a surface
So if the violent movement was due to sliding, skid is probably the correct word to use.
If you skid in particular ways, there may be more specific terms for the action, like fishtail or spin out.
to have the rear end slide from side to side out of control while moving forward
a rotational skid by an automobile that usually causes it to leave the roadway
Swerve refers to a sudden change of direction that the driver initiated, rather than due to the car losing control.
to turn aside abruptly from a straight line or course
Swerve is commonly used to refer to suddenly changing direction to avoid hitting something in the direction you were traveling, that the brakes could not have stopped you in time for. It can also be used in cases where the driver was simply careless, or otherwise compromised (drunken driving will often be described as swerving).
In the context of driving a car, drift can be used in two very different ways. Which is meant depends on context.
The first is the car drifting with respect to its intended trajectory, which are generally the lanes on the road. This sense of the word is not necessarily specific to driving, but it is often used for driving.
If someone or something drifts into a situation, they get into that situation in a way that is not planned or controlled.
This is a gradual effect, from the car not being precisely aligned with the lanes, and is common with impaired drivers (inebriated, or over-tired, or simply distracted). May result in swerving to right oneself, after drifting away from where you should be.
Drifting out of lane and swerving back is the hallmark of an impaired driver, and precisely what police look for when pulling over suspected drunk drivers. The entire drift-and-swerve-back style is often described as erratic. Erratic driving is very dangerous, because it both means that the car will not be where it should be (and collisions, by definition, require at least one person to be where they shouldn’t be, since collisions shouldn’t happen), and because it is extremely difficult for other drivers to predict. It can also have cascading effects: someone drifting and swerving forces other drivers to react, maybe forcing them to swerve themselves, amplifying the danger significantly. Responsible drivers recognize erratic driving and leave a lot of empty road between them to avoid this.
The second meaning of drift with respect to driving a car is an intentional skid, used in the context of racing.
Drifting is a driving technique where the driver intentionally oversteers, with loss of traction in the rear wheels or all tires, while maintaining control and driving the car through the entirety of a corner.
Since we’re talking about racing, this usage of drift is associated with driving at very high speed. It is also a very difficult and dangerous maneuver; no one should be using it on a public road (but then, street racing is a thing, so people do—The Fast and the Furious: Tokyo Drift is a movie literally named after using the maneuver in illegal street racing).
None of these imply a collision
Skidding, including fishtailing or spinning out, can be recovered from without hitting anything (in a sufficiently open environment, there may well not be anything to hit). Swerving is a dangerous maneuver that can make things worse, but it doesn’t automatically do so—if there is room, it can avoid a collision instead (the big problem is that the sudden nature of a swerve often prevents the driver from taking the time to ensure that there is room). The first sense of drifting is rarely, in and of itself, a problem—usually it’s the symptom of a bigger problem (being impaired and prevented from driving properly). And in the second sense, in the context of a race and a driver with sufficient skill, a drift can be a common and perfectly safe maneuver.
So any of these explain why a collision happened, but if you just say that someone skid, swerved, or drifted, we don’t know that a collision occurred. You would have to use “and crashed,” as you do in your question.