Does a character know that he is cursed while he is under the effect of bestow curse?
The answer would be interesting in order to know if in general the target of bestow curse will try to remove or identify the curse.
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A target would definitely be aware that they're under some sort of negative effect once they spend a little time in combat. Whether they have disadvantage on attacks, waste turns, or take more damage than they should, they would quickly realize something was amiss. These effects are all very noticeable, so even a round or two is combat should be sufficient to notice an effect.
Additionally, if they noticed the spell being cast and identified the spell at that time, they would obviously know they're under the effects of the spell.
Where it gets murky is if the spell is cast on them but they haven't been in combat yet. At that point, there are no other perceptible effects, so it's most likely they wouldn't be aware of the spell without something like Detect Magic.
It's also worth noting that even in cases where they realize they're under some sort of negative effect, they wouldn't necessarily know that it was magical, or what effect it was. They could very reasonably assume they'd been poisoned or were falling ill with some kind of disease. Whether they connect the effects they're experiencing with a magical effect would depend on several factors, especially including whether or not they saw the spell being cast on them.
This might be a good time for an Arcana check - on a very low check, they might not realize it's magical at all. On a low to moderate check, they could realize it is a magical effect but not know the specific spell (this would probably be typical of most enemies). On a high check, they could identify the spell. If they do realize it's a magical effect, it's reasonable that they would seek help to dispel it if such help was available.
From the Player's Handbook p. 204
Unless a spell has a perceptible effect, a creature might not know it was targeted by a spell at all. An effect like crackling lightning is obvious, but a more subtle effect, such as an attempt to read a creature’s thoughts, typically goes unnoticed, unless a spell says otherwise.
As such, the target of bestow curse does not know the spell was cast on it unless it identified the spell when it was cast (such as via the variant spell identification from Xanathar's Guide to Everything).
Contrast this to the spell charm person which specifically says
When the spell ends it knows it has been charmed.
However, in some cases the curse may have a noticeable effect. If one of the curses has a perceivable effect (such as one of the alternate effects that a DM can approve), the character may be able to discern, likely through an Arcana check, that he/she is cursed.