Although in other countries, especially outside of Europe, the "Order of the Garter" is often introduced as extremely prestigious, for example, claiming that the order of knights "originated in the Middle Ages"; it is "the oldest knightly order in the world today"; and it's "the United Kingdom's highest and oldest order of chivalry," but none of this is true.
Firstly, the Order of the Garter wasn't established until the middle of the fourteenth century (generally believed to be 1348); for this reason, it never had the chance to participate in the Crusades.
From a certain perspective, this order of knights wasn't so much established by the British in adherence to medieval traditions, as it was hastily assembled to keep up with European fashion.
As for the choice of the garter as the symbol of the order, it was likely because medieval knights often used garters to strap their weapons tightly to their bodies, hence it became a symbol of valor and combat prowess.