A belt should hold that down so that it's not bouncing at all. It's worth finding one that's comfortable, and you can get used to it pretty quickly.

I didn't like mine touching my skin so I would tuck in my shirt while running with a belt.

For a bigger ergonomics/comfort challenge, you might even extend out to run lengths where you have to bring your own water. I switched between handheld water bottles and backpacks, depending on length of run and amount of water needed. At that point, the weight and size of a phone is pretty insignificant in comparison.

I don't run as much anymore, but these days I'm on a smartwatch plus wireless headphones. The watch doubles as a payment card too, in case I want to buy something while I'm out. Like candy fuel for my run.