Recent advances in the development of non-fullerene acceptors have increased the power conversion efficiency of organic solar cells to approximately 13%. Fullerene-derivatives and non-fullerene acceptors possess distinctively different structural, optical and electronic properties, which also change the requirements on the polymer donor in non-fullerene organic solar cells. Therefore, in this study, the effect of the molecular weight of the conjugated polymer on the photovoltaic performance, charge carrier mobility, crystallization properties, film morphology, and non-geminate recombination dynamics is systematically investigated in polymer:small molecule organic solar cells using the low bandgap polymer PTB7-Th as the donor and the non-fullerene indacenodithiophene-based small molecule O-IDTBR as the acceptor. Among the examined polymer samples (50–300 kDa), high molecular weights of PTB7-Th (with an optimum molecular weight of 200 kDa) are advantageous to achieve high efficiencies up to 10%, which can be correlated with an increased crystallinity, an improved field-effect hole mobility (1.05 × 10−2 cm2 V−1 s−1), lower charge carrier trapping and a reduced activation energy of charge transport (98 meV). Bias-assisted charge extraction and transient photovoltage measurements reveal higher carrier concentrations (1016 cm−3) and long lifetimes (4.5 μs) as well as lower non-geminate recombination rate constants in the corresponding devices, supporting the high photocurrents (ca. 15.2 mA cm−2) and fill factors (>60%).