Breathless (1983) on Netflix

Jesse Lujack - only one of the many aliases by which he goes - is a small time Las Vegas hood, who nonetheless believes that the world should figuratively bow at his feet. He steals a car to drive to Los Angeles to convince Monica Poiccard, a French co-ed with who he has only a casual relationship but who he professes to love, to run off to Mexico with him. Along the way, he is pursued by highway patrol due to a traffic violation. Stopped, Jesse flagrantly shoots and critically wounds the police officer. This incident makes his getaway to Mexico all the more important. Monica is torn by his offer: she is attracted to him and his confidant and dangerous ways (unaware of the incident en route to Los Angeles), yet she is on the verge of a relationship with one of her professors who would be able to advance her career. Jesse's time in Los Angeles too is torn in two ways: he wants to convince Monica of his love for her and thus why she should run off with him, but his need to get away is affected by the facts that the authorities know he shot the police officer, that police officer eventually died making him a cop killer, and his name and face are plastered all over the media. Monica may come to her own decisions based on her feelings for Jesse and eventually finding out about the police shooting.