Flexible dielectric polymers that can withstand high electric field and simultaneously have high dielectric constant are desired for high-density energy storage. Here, we systematically investigated the impact of oxygen-containing ether and carbonyl groups in the backbone structure on dielectric properties of a series of cyclic olefin. In comparison to the influence of the –CF₃ pendant groups that had more impact on the dielectric constant rather than the band gap, the change of the backbone structure affected both the dielectric constant and band gaps. The one polymer with ether and carbonyl groups in the backbone has the largest band gap and highest discharge efficiency, while it has the lowest dielectric constant. The polymer without any ether groups in the backbone has the smallest band gap and lowest discharge efficiency, but it has the highest dielectric constant. Polymers that have no dipolar relaxation exhibit an inversely correlated dielectric constant and band gap. Enhancing the dipolar relaxation through rational molecular structure design can be a novel way to break through the exclusive constraint of dielectric constant and band gap for high-density energy storage.