So Whitebeard probably ditched the Rocks Pirates fight 'cause it was a lost cause. Kaidou probably wanted to fight Rocks but got his butt kicked. So, Linlin felt bad and saved him, which made him owe her.
Garp and Roger teamed up to take him down, but that was before Saturn/Imu gave Roger that incurable disease.
Still doesnt explain why Imu didn't order Garps death. Oda lost the plot with this one.