Understanding microscopic parameters that control steepness of the temperature variations of segmental relaxation (fragility) and the glass transition phenomenon remains a challenge. We present dielectric and mechanical relaxation studies of segmental dynamics in various polymers with different side groups and backbone structures. The results have been analyzed in terms of flexibility of backbone and side groups of polymeric molecules, as suggested by the recent theoretical works by Dudowicz et al. A comparison of structures with identical backbones and varying side groups and identical side groups but different backbones reveals that the flexibility of side groups relative to the flexibility of the backbone is the most important factor controlling fragility in polymers, while the glass transition temperature T g depends primarily on the backbone flexibility and the side group bulkiness (occupied volume). Based on these results and analysis of literature data we formulated a modified approach to understand the role of chemical structure in segmental dynamics: (i) Polymers with stiff backbones always have high T g and fragility, while (ii) polymers with flexible backbones and no side groups are the strongest; (iii) however, for the most common type of polymeric structure, C-C or Si-O backbone with side groups, fragility increases with increasing "relatiVe" stiffness of side groups versus the backbone. In this class of polymers, lowest fragility is expected when the side groups are of similar chemical structure (or flexibility) as the backbone, as in the case of polyisobutylene, one of the strongest polymers known.