The Fort save v.s. being sickened is at the end of each of your opponent's turns, but the Ref save v.s. being entangled is unclear. It certainly occurs at the end of your turn after casting the spell, but since it says 'your turn' and not 'your turns' or 'each of your turns while the duration persists' it's unclear whether or not they need to make a save on each additional turn you end. Because of the ordering of the text, it's also unclear whether creatures succeeding on a save for entering the area would need to also save against a turn ending effect. To make this a little clearer, I'll go over a typical casting:
1- You cast the spell. Every creature in the affected area must immediately save or become entangled. Those that fail are entangled until they break free or the spell ends (leaving the affected area does not remove the condition).
2- You end the turn on which you cast the spell. Probably noone's moved, so anyone who made their save the first time now has to save again. Those that failed don't have to make a save, even if they've somehow broken free. Those that fail this save are entangled just like those that failed the first save, except they might be subject to additional saving throws on future turns.
3- Other creatures take actions on their turns. Any that enter the affected area must save or gain the entangled condition, and make a separate save or gain the sickened condition. If they succeed at their save they might have to save again at the end of your next turn v.s. entanglement if they remain in the area. Regardless of success or failure, they will definitely have to save again v.s. sickening at the end of their turn if they remain in the area. Also, any creature who starts and ends their turn in the area has to save v.s. sickening (once), but not entanglement.
4- Your next turn starts. You do stuff, and no one has to save v.s. entanglement except maybe you.
5- Your next turn ends. Many people might possibly need to make a save now, but no one definitely does (except maybe you).
Now, your DM has indicated they are going with the reasonable interpretation that 'your turn' refers exclusively to the first turn after you cast the spell. That means:
1- You cast the spell. Every creature in the affected area must immediately save or become entangled. Those that fail are entangled until they break free or the spell ends (leaving the affected area does not remove the condition).
2- You end the turn on which you cast the spell. Probably noone's moved, so anyone who made their save the first time now has to save again. Those that failed don't have to make a save, even if they've somehow broken free. Those that fail this save are entangled just like those that failed the first save.
3- Other creatures take actions on their turns. Any that enter the affected area must save or gain the entangled condition, and make a separate save or gain the sickened condition. Regardless of success or failure, they will definitely have to save again v.s. sickening at the end of their turn if they remain in the area. Also, any creature who starts and ends their turn in the area has to save v.s. sickening (once), but not entanglement.
4- Your next turn starts. You do stuff, and no one has to save v.s. entanglement except you if you walk into the area.
5- Your next turn ends. No one needs to make a save now, except you if you ended your turn in the affected area.
repeat 3, 4, and 5 until the spell wears off.