Astral Heron says if you play your first card while its at a battlefield, your next card is discounted. but it does not say 'this turn' so if you end turn, is your next card discounted still? why?

No — the discount does not carry over. Astral Heron was errata'd (2026-07-24) so its current text reads "the next card you play this turn costs [2][A][A] less."

The "this turn" qualifier means the discount is a Delayed Passive Ability whose window ends in the Expiration Step, where all "this turn" effects expire simultaneously (Rule 317.2.c). If you end your turn without playing another card, the discount is gone — it will not apply to your first card next turn.

Your premise matches the old text, which lacked "this turn" and was ambiguous — but that ambiguity is exactly what the errata resolved: the discount was always intended to last only until the end of the turn it was created.