Yeah- it's normal (especially if you aren't reading signals, have not trained "step up" fully and/or keep accidentally breaking trust by pushing things to where you end up getting bitten).
They move slowly.
Build trust and never try to move faster than they are ready.
You want your interactions to be as low-key and positive as possible---biting is a last-resort behavior in the wild...and so if yours keeps doing it, you may have accidentally taught it that the only way to assert its desire to stay put is to bite (or you will keep at it).
It's all stuff that can be fixed, but much of this depends on what you have been doing.
I agree with much of what wrench said, but if you shun a bird that bites in the cage, you could be rewarding the behavior. Shunning depends on the bird WANTING your company. He cannot pick up a bird who bites him in protest of stepping up (that would really mess with trust) so he can't put the bird in a neutral place. If the bird bites and he walks off and leaves the bird in the cage, that may be exactly what the bird wanted, but there is no way to push it past that point and force to bird to do something without REALLY setting yourself back.
This is why it is absolutely essential that you read signals better, and avoid getting bitten.
Outside of the cage it's way easier to shun a bird...but even then, you have to be sure that they are not biting as a means of getting away from your company. A few years ago, Noodles bit me so I took her to her cage, but then she started biting in the evening to go to her cage..SO, just think about the patterns..
Almost all behaviors fall into escape (from someone, a situation, a stimulus, an undesired task), attention (yelling, eye contact, verbal, physical reactions etc), sensory (eating when hungry, crying when injured etc), and tangibles (to get a treat, physical reward, toy etc).
It sounds like your bird is biting to escape the aversive task of stepping up. In the event that a behavior is escape oriented, shunning doesn't work as well because it kind of rewards the behavior...at least when it happens while the bird is perched (because trying to forcefully move a bird who is showing you NO with its beak will only increase your bird's biting severity and familiarity, while harming the relationship).
You need to find out what your bird loves and catch it being good. Pair key phrases with actions and rewards **not just tangibles--whatever your bird loves**...move slowly...try target training etc..Keep trainings brief and end on a positive without pushing too hard. REMEMBER--- If I am doing a behavior to get away from something....say, a test (e.g., ESCAPE), no amount of candy (tangible) is going to change my mind. To reinforce a behavior you want to see more of, you have to reward with a reinforcer that matches the function of the behavior. A kid who is acting out in class for attention gets yelled at by the teacher (that is actually reinforcing the attention seeking behavior b/c the kid did it for attention, and got attention)...You will not get an attention seeker to stop seeking attention, so you have to teach them more positive ways to do it. If a kid is yelling out, you teach them a signal and let them talk ASAP...see where I am going with this?
You need to read signals and build trust so that your bird WANTS to step up. In the meantime, take baby steps and try to reward as much as possible for tiny approximations towards that goal of step up. NOTE: YOU CAN ABSOLUUUUUUTELY pair any behavior with additional rewards (food etc) but what I meant above was that the initial/most basic function behind any behavior (attention, escape, sensory or tangibles) must be considered first-- just b/c it seems like a reward to you, doesnt mean your bird will agree if it isn't motivating to him/her. If your bird loves attention, you can still give that attention and a treat, BUT a treat alone, with little attention would be less motivating for an attention-seeking bird. Look up ABA (applied behavior analysis) if you want to research what I am talking about more.
Almost forgot- if you have huts, tents, boxes etc in the cage (or if the bird has a lot of access out of the cage) remove them. These can lead to hormonal behavior and huts are very dangerous for other reasons as well. Make sure you only ever pet on the head and neck (the rest is sexual) and make sure your bird gets 10-12 hours of sleep nightly. 10 is decent for a conure, but if you don't have a sleep schedule, you will have a crankier bird..Sleep needs to be set (just like it is in the wild). Sleep regulates their hormonal and immune health (and impacts mood as well)
