Tailor-made materials featuring large tunability in their thermal transport properties are highly sought-after for diverse applications. However, achieving `user-defined' thermal transport in a single class of material system with tunability across a wide range of thermal conductivity values requires a thorough understanding of the structure-property relationships, which has proven to be challenging. Herein, large-scale computational screening of covalent organic frameworks (COFs) for thermal conductivity is performed, providing a comprehensive understanding of their structure-property relationships by leveraging systematic atomistic simulations of 10,750 COFs with 651 distinct organic linkers. Through the data-driven approach, it is shown that by strategic modulation of their chemical and structural features, the thermal conductivity can be tuned from ultralow (≈0.02 W m-1 K-1) to exceptionally high (≈50 W m-1 K-1) values. It is revealed that achieving high thermal conductivity in COFs requires their assembly through carbon-carbon linkages with densities greater than 500 kg m-3, nominal void fractions (in the range of ≈0.6-0.9) and highly aligned polymeric chains along the heat flow direction. Following these criteria, it is shown that these flexible polymeric materials can possess exceptionally high thermal conductivities, on par with several fully dense inorganic materials. As such, the work reveals that COFs mark a new regime of materials design that combines high thermal conductivities with lowdensities.