Why did Gatsby switch cars?

From what I can see, your answer lies in Chapter Seven. In this chapter, Daisy suggests that everyone should go into town.

Meanwhile, the atmosphere in Tom and Daisy's house is charged with a percolating tension that threatens to erupt into outward conflict. At lunch, Daisy is brazenly affectionate with Gatsby; she doesn't bother to hide the fact that she's infatuated with her lover.

For his part, Tom senses that Daisy is past all pretense when it comes to Gatsby, and he's definitely angry about being so obviously cuckolded. It's Tom who suggests that he and Gatsby switch cars. Meanwhile, Gatsby isn't too keen on the idea; he's just a tad uncomfortable with the idea of driving his lover's husband's car. Gatsby finds himself in a difficult position; if he objects, Tom is the kind of man who has no qualms about making an embarrassing scene. If Gatsby consents, he's playing into Tom's hands and publicly consenting to being emasculated in front of Daisy.

Since it's a no-win situation for Gatsby, he decides to cut his losses and to let sleeping dogs lie. He reluctantly consents to let Tom drive his car, while Tom drives his. For his part, Tom feels a need to exert himself in order to salvage what masculine pride he has left. After bullying his way through the exchange, Tom, calling his rival's car a "circus wagon," attempts to get Daisy to ride with him in Gatsby's car. However, Daisy evades her husband's grasp and decides to ride with Gatsby in Tom's car.

Daisy leaves Tom aghast and furious at her blatant disrespect. It's obvious that Tom meant to embarrass Gatsby, but the tables have now been turned on him. So, in truth, Gatsby only consented to switch cars with Tom because the latter pressured him into doing so.