We investigate the effects of fragmentation in equity markets on the quality of trading outcomes in a panel of FTSE stocks over the period 2008-2011. This period coincided with a great deal of turbulence in the UK equity markets, which had multiple causes that need to be controlled for. To achieve this, we use the common correlated effects estimator for large heterogeneous panels. We extend this estimator to quantile regression to analyse the whole conditional distribution of market quality. We find that both fragmentation in visible order books and dark trading that is offered outside the visible order book lower volatility. But dark trading increases the variability of volatility, while visible fragmentation has the opposite effect, in particular at the upper quantiles of the conditional distribution. The transition from a monopolistic to a fragmented market is non-monotonic with respect to the degree of fragmentation.