Supercooled liquids display sluggish dynamics, often attributed to their structural characteristics, yet the underlying mechanism remains elusive. Here we conduct numerical investigations into the structure-dynamics relationship in model glass-forming liquids, with a specific focus on an elementary particle rearrangement mode known as the 'T1 process'. We discover that the ability of a T1 process to preserve glassy structural order before and after is pivotal towards determining a liquid's fragility-whether it exhibits super-Arrhenius-like or Arrhenius-like behaviour. If a T1 process disrupts local structural order, it must occur independently without cooperativity, resulting in Arrhenius-like behaviour. By contrast, if it can maintain order, it sequentially propagates from disordered peripheries to the middle of high-structural-order regions, leading to cooperativity and super-Arrhenius-like behaviour. Our study establishes a microscopic link between liquid-structure ordering, dynamic cooperativity and super-Arrhenius-like dynamics, extending the understanding of the structure-dynamics relationships in supercooled liquids.