The skew and shape of the molecular weight distribution (MWD) of polymers have a significant impact on polymer physical properties. Standard summary metrics statistically derived from the MWD only provide an incomplete picture of the polymer MWD. Machine learning (ML) methods coupled with high-throughput experimentation (HTE) could potentially allow for the prediction of the entire polymer MWD without information loss. In our work, we demonstrate a computer-controlled HTE platform that is able to run up to 8 unique variable conditions in parallel for the free radical polymerization of styrene. The segmented-flow HTE system was equipped with an inline Raman spectrometer and offline size exclusion chromatography (SEC) to obtain time-dependent conversion and MWD, respectively. Using ML forward models, we first predict monomer conversion, intrinsically learning varying polymerization kinetics that change for each experimental condition. In addition, we predict entire MWDs including the skew and shape as well as SHAP analysis to interpret the dependence on reagent concentrations and reaction time. We then used a transfer learning approach to use the data from our high-throughput flow reactor to predict batch polymerization MWDs with only three additional data points. Overall, we demonstrate that the combination of HTE and ML provides a high level of predictive accuracy in determining polymerization outcomes. Transfer learning can allow exploration outside existing parameter spaces efficiently, providing polymer chemists with the ability to target the synthesis of polymers with desired properties.