There is a stream delay setting in OBS — check your settings on the source setup running OBS. 5 seconds sounds like the default, you could potentially set it to 0. This is in place to

a) head off any network dropped frames and allow OBS to backfill the stream buffer. At 0 seconds you may see slightly more quality drop from 1080p to 480p if your stream source upload bandwidth is unreliable than if you left this at 5+ seconds.

b) for game streamers of competitive multiplayer video games, to protect them from "stream cheating" where an opponent recognizes their in-game ID and then looks at their stream to get additional information in the middle of a match (eg: looking at their hand in poker). With many games a 5 second delay nullifies this issue, and many streamers set it to more like 30-45 seconds depending on the game they play.