Upvote:2
I'll add some metallurgy information to complement the answers.
There are several different kinds of steel. The basic one, which is iron and carbon, has different behaviors depending of the % of carbon in the alloy. Few carbon and you have mostly iron, which is soft and malleable, while a lot of carbon means that the steel will be hard but fragile. Since is difficult to achieve the correct amount of carbon in an steel alloy, or to remove all carbon to have pure iron, not always the steel is better than bronze. Therefore, depending of the quality of the smith, bronze would be still competitive against a bad iron/steel.
Upvote:4
Possibly the Battle of Kadesh. Check out this video.
Upvote:7
Also check out Qin Shi Huang, the first emperor to unify China (same man who's tomb has the terracotta army surrounding it). The Qin province sent forth massive armies using bronze while a lot of the rest of China was transitioning to iron/steel (and Qin won in part due to their high grade of bronze compared to low the grade iron weapons of the time, for example Qin bronze swords were able to be made at longer lengths than their contemporaries). So right there is a ton of battles (some of which fielded over half a million soldiers in a single battle).