The White Castle · Board game rules
When does the gardener place dice on the bridge?
Quick answer
Gardeners don't place dice on Bridges — dice are placed by any player choosing from a Bridge; Gardeners are activated separately at the end of a round if dice remain on their Bridge.
Details
There is no rule allowing a Gardener piece to place dice on a Bridge. Dice are placed by players during their turn by choosing a die from one of the two ends of a Bridge and moving it onto a space on the main board or Domain board [p.4 — Using the Dice].
The Gardener action itself involves taking your leftmost Gardener piece and assigning it to a Garden card (paying Food), then carrying out the action on that card — this has nothing to do with placing dice on Bridges [p.10 — Sending Clan Members to the Court of Himeji].
What may be causing confusion is the End of Round step: if, when the round ends, there are still dice remaining on a Bridge, the Gardeners assigned to the Garden card below that Bridge are "activated," letting each player use their Gardener's card action once, in turn order [p.13 — End of the Round].
Watch out: This activation only happens if at least 1 die is still sitting on the Bridge above the Garden card when the round ends — it does not involve Gardeners placing dice themselves. If your question was about something else (e.g., how dice are set up on Bridges at the start of a round), let me know and I can clarify further [p.13 — End of the Round].