Why Karna’s fight happened with king Jarasandha?
"Narada said, 'Hearing of the fame1 of Karna's might, the ruler of the Magadhas, king Jarasandha, challenged him to a single combat.
1: Karna defeated many kings in a Svayamvar, more details can be found in
The Mahabharata, Book 12: Santi Parva: Rajadharmanusasana Parva: Section IV.
Details of the combat
A fierce battle took place between Karna and Jarashandha in which they struck each other with various kinds of celestial weapons.
At last when their arrows were exhausted and bows and swords were broken and they both became carless, they began, possessed of might as they were, to fight with bare arms.
While engaged with Jarasandh in mortal combat with bare arms, Karna was about to sever the two portions of his antagonist's body that had been united2 together by Jara.
2 : Details on this can be found in The Mahabharata, Book 2: Sabha Parva: Rajasuyarambha Parva: Section XVII.
How the combat came to an end
The king (of Magadha), then after feeling himself very much pained, cast off all desire of hostility and addressed Karna, saying, 'I am gratified.' From friendship he then gave unto Karna the town Malini. Thus Karna became famous on earth for the valour of his arms. ~The Mahabharata, Book 12: Santi Parva: Rajadharmanusasana Parva: Section V