Did Joshua and the Israelites capture Jerusalem?

Answer: In Joshua 10, it is the king of Jerusalem that is killed outside Jerusalem (verses 16-18 and 22-26). The city itself was not captured. The five kings of Jerusalem, Hebron, Jarmuth, Lachish, and Eglon, went out to war near Gibeon (Joshua 10:5). They lost the battle (v13-14), hid in a cave at Makkedah (v16), and were eventually killed (v26). Joshua did not siege and capture Jerusalem, as chapter 15 says.

Conclusion: No contradictions. The king of Jerusalem was killed in a battle outside Jerusalem but the city itself was not captured.