From what I understand, evasive and stunt driving courses usually run at around two to four weeks. Now, that is simply to learn what you’re supposed to do. Not to hone it to a professional level.
When you’re watching films, you’re usually looking at people who’ve spent ten to fifteen years doing utterly insane things behind the wheel. Not something you’re going to fully pickup in under a month. The basics and principles? Sure, but not that level of proficiency.
As for what training actually entails, I don’t know much more than you. My understanding is that Secret Service and DSS training starts the cadets in sports cars, and gradually works them down to vehicles with inferior handling, until they can perform high speed maneuvers in an SUV. However, I don’t know how widespread that approach is.
Depending on where you’re living, stunt driving courses are available, though they’re not particularly cheap, as I recall. Also, depending on what country you’re in, basic driver training required to get a license may be significantly more demanding than in the US. Pretty much anyone with a law enforcement background will have received some specialized driver training. Police prefer high power, rear wheel drive cars, which give a performance edge over most vehicles on the street, but also require more skill to operate, particularly at high speeds.
You’re right to question characters suddenly demonstrating that kind of driving skill. It does require training and practice. It is something a character can know, without the people around them realizing it. But it’s not something they “just know,” without explanation.