This paper critically assesses the effect of fossil fuel dependence and polluting emissions from the transport sector on the performance of logistics operations in the context of Green Supply Chain Management (GSCM). We collected macro-level time-series data for a sample of 27 European Union (EU) countries over the period 2007–2018. A new Artificial Neural Networks (ANNs) algorithm is adopted in a multivariate framework to investigate the dynamic interactions among a range of Logistics Performance Indexes (LPI), the demand for oil products, and carbon dioxide (CO2) emissions from fuel combustion in the transport sector. Empirical findings show that oil product consumption and CO2 emissions sharply influence the transport logistics indexes. However, a feedback relationship is discovered for environmental pollution, indicating that oil use is not significantly driven by supply chain performance. Based on our empirical insights, adequate policy recommendations are provided to help turning the logistics sector towards a more sustainable path in the European area.