In Brawl, turn order is mainly decided by the Speed stat. The Speed stat will vary depending on the dragon type. As a reminder, the dragon types are: 

• Cunning: Purple icon, have better special attacks.
• Swift: Blue icon, have better speed.
• Furious: Red icon, have better attacks.
• Protector: Green icon, have better health + defense.

As Speed depends on the dragon type, this means some dragons may be faster than others. How does this translate to turn order? Faster dragons' turns arrive faster than those that are slower.

 

Swift dragons being the fastest is the best example, as these dragons are so fast they may have more than one turn before you get to the top of the round again. If your opponents have some Swift dragons in their team, that means they will have more turns.