Whenever Thor goes from place to place magically a strange design appears in the ground under him. But in The Avengers, Thor & Loki went to Asgard. But no design appeared under them? Why didn't it happen?
Because they used Tesseract/Space stone's powers:
Both rotate the handle of Tesseract case and teleported with blue light.
Also we already know biforst was not in working condition during the film events:
LOKI: Oh, you should thank me. With the Bifrost gone how much dark energy did the Allfather have to muster to conjure you here? Your precious Earth. - source
Those "designs" you mentioned, are created when the "bifrost" or any bifrost related mechanism/tool is used for teleportation. The signature pattern is also noted when Thor arrives in Wakanda using the teleportation power of StormBreaker.
In the instance in question, they used the space stone instead to teleport, hence the absence of the pattern. This is supported by the absence of that pattern too in the first instance of teleportation by the space stone (Red Skull) in Captain America: The First Avenger.