The damage can be applied more then twice a round though.
Turn 1 It is intially cast, doing damage.
Turn 2 the creature's turn; it does not move out. Maybe it is afraid of OA from nearby fighters. bonfire does damage.
Turn 3 a player push the creature out of the bonfire.
Turn 4 a player pushes the creature back into the fire. (Example, by using thorn whip). Bonfire does damage . The spell says
A creature must also make the saving throw when it enters the bonfire’s space for the first time on a turn or ends its turn there.
Turn 5 A player first push the creature out, and then use a multiattack or something to drag it back in. Bonfire does damage.
In this round it has then done damage 4 times.
In this scenario it is strong, but dependent on your other players to work with it, it is unlikely that it is able do damage that many times.
This is a digression, but similar mechanics work with moonbeam:
Moonbeam is stronger since it does damage at the start, not the end of a creatures turn. So the creature can't flee out of the effect on its turn. You can redirect the moonbeam using your action, so it's essentially the same as casting a new bonfire if the creature moves out of it.
Also the damage is not applied initially, only on the creatures turn so you can't use the movement of moonfire to move it back and forth in to a creature, or scorching many creatures during it's movement.
But you can do the same tactic as bonfire with pushing and pulling since the spell says " enters the spell’s area for the first time on a turn or starts its turn there "
this is confirmed in a sage article. https://dnd.wizards.com/articles/features/rules-answers-april-2016
So I assume the same applies to bonfire since it is worded the same.
I'm not sure what would happen if someone pushed a creature through a moonbeam or bonfire, not into it since they do enter it, but only for a very short time, and does not land there. I guess it is up to the GM?
Flaming sphere works similar to bonfire, doing damage at the end of a creatures turn, not the start. But it does not do damage initially, unless you use your bonus action to move it, then ram it in to a creature, but then it stops when you hit a creature (so you can't scorch it over several creatures) and it doesn't do damage when a creature enters it, so you can run through it or be pushed through it.
All of these spells can damage more then once per round. Bonfire and flaming sphere are really good if placed where movement, or OA, is an issue for a monster. Bonfire and moonbeam are better if the whole party push enemies around.