Another common expression is: éclater en sanglots
Elle éclata en sanglots.
Literally this means to burst into sobbing, so in that respect, it`s not exactly the same as bursting into tears. However, it does capture the (somewhat violent) notion of "bursting" (éclater), which "fondre en larmes" misses completely.
Fondre en larmes literally means "to melt into tears" and would probably not be used if someone all of a sudden started loud weeping. In that case "éclater en sanglots" could be more appropriate.
When I hear "fondre en larmes" it makes me think of someone staring out at the sea, and starting to cry noiselessly thinking about their long lost home (or something like that). You would hardly say "burst into tears" for that!