According to the Secret Histories, Temujin (or rather Borte who told Temujin) suspected Jamuka was conspiring against him. So Temujin abandoned him and led (fled) his followers away from Jamuka's camp in the night.

They might have left it at that but some of Jamuka's followers abandoned him in favour of Temujin. The Secret Histories mention they favoured Temujin because oh various omens, one of them being that an ox kept saying "Heaven and Earth agree, let Temüjin be the nation’s master.” (Yeah...) Anyway they altogether proclaimed Temujin as Genghis Khan.

Their relation was strained but they became enemies when one of Temujin's men killed Taichar, Jamuka's brother for stealing some animals. Jamuka raised an army and went after Temujin. Temujin raised his own men and met him at the Onon river but was defeated and fled. Jamuqa boiled alive some of Temujin's captured men. They believed boiling a man destroyed the spirit, preventing it from entering the afterlife so this was a pretty serious offence.