Identifying causal relationships and quantifying their strength fromobservational time series data are key problems in disciplines deal-ing with complex dynamical systems such as the Earth system orthe human body. Data-driven causal inference in such systems ischallenging since datasets are often high-dimensional and nonlinear with limited sample sizes. Here we introduce a novel method thatflexibly combines linear or nonlinear conditional independence testswith a causal discovery algorithm to estimate causal networks fromlarge-scale time series datasets. We validate the method on timeseries of well-understood physical mechanisms in the climate sys-tem and the human heart and using large-scale synthetic datasetsmimicking the typical properties of real world data. The experi-ments demonstrate that our method outperforms state-of-the-arttechniques in detection power, which opens up entirely new possi-bilities to discover and quantify causal networks from time seriesacross a range of research fields.