There is missing context for the scene from The Expanse. Alex (the pilot) gets the computer to plot a course using manoeuvring thrusters only, so it is not just a series of gravity assists, there is also manoeuvring to adjust course.
I would say that one of two things (or both) is happening:
- Alex is leaning because of the manoeuvring thrusters firing
- He is leaning like someone trying to steer harder in a video game.
Also, he drinks a fair few beers while hiding behind Cyllene before this, so that would support 2 I think.