We present an algorithm for phylogenetic reconstruction using quartets that returns the correct topology for $n$ taxa in $O(n \log n)$ time with high probability, in a probabilistic model where a quartet is not consistent with the true topology of the tree with constant probability, independent of other quartets. Our incremental algorithm relies upon a search tree structure for the phylogeny that is balanced, with high probability, no matter what the true topology is. Our experimental results show that our method is comparable in runtime to the fastest heuristics, while still offering consistency guarantees.