This is quite a ubiquitous misunderstanding of the central limit theorem, which I have also encountered in my statistical teaching. Over the years I have encountered this problem so often that I have developed a Socratic method to deal with it. I identify a student that has accepted this idea and then engage the student to tease out what this would logically imply. It is fairly simple to get to the reductio ad absurdum of the false version of the theorem, which is that every sequence of IID random variables has a normal distribution. A typical conversation would go something like this.
Teacher: I noticed in this assignment question that you said that because $n$ is large, the data are approximately normally distributed. Can you take me through your reasoning for that bit?
Student: Is that wrong?
Teacher: I don't know. Let's have a look at it.
Student: Well, I used that theorem you talked about in class; that main one you mentioned a bunch of times. I forget the name.
Teacher: The central limit theorem?
Student: Yeah, the central limit theorem.
Teacher: Great, and when does that theorem apply?
Student: I think if the variables are IID.
Teacher: And have finite variance.
Student: Yeah, and finite variance.
Teacher: Okay, so the random variables have some fixed distribution with finite variance, is that right?
Teacher: And the distribution isn't changing or anything?
Student: No, they're IID with a fixed distribution.
Teacher: Okay great, so let me see if I can state the theorem. The central limit theorem says that if you have an IID sequence of random variables with finite variance, and you take a sample of $n$ of them, then as that sample size $n$ gets large the distribution of the random variables converges to a normal distribution. Is that right?
Student: Yeah, I think so.
Teacher: Okay great, so let's think about what that would mean. Suppose I have a sequence like that. If I take say, a thousand sample values, what is the distribution of those random variables?
Student: It's approximately a normal distribution.
Teacher: How close?
Student: Pretty close I think.
Teacher: Okay, what if I take a billion sample values. How close now?
Student: Really close I'd say.
Teacher: And if we have a sequence of these things, then in theory we can take $n$ as high as we want can't we? So we can make the distribution as close to a normal distribution as we want.
Teacher: So let's say we take $n$ big enough that we're happy to say that the random variables basically have a normal distribution. And that's a fixed distribution right?
Teacher: And they're IID right? These random variables are IID?
Student: Yeah, they're IID.
Teacher: Okay, so they all have the same distribution.
Teacher: Okay, so that means the first value in the sequence, it also has a normal distribution. Is that right?
Student: Yeah. I mean, it's an approximation, but yeah, if $n$ is really large then it effectively has a normal distribution.
Teacher: Okay great. And so does the second value in the sequence, and so on, right?
Teacher: Okay, so really, as soon as we started sampling, we were already getting values that are essentially normal distributed. We didn't really need to wait until $n$ gets large before that started happening.
Student: Hmmm. I'm not sure. That sounds wrong. The theorem says you need a large $n$, so I guess I think you can't apply it if you only sampled a small number of values.
Teacher: Okay, so let's say we are sampling a billion values. Then we have large $n$. And we've established that this means that the first few random variables in the sequence are normally distributed, to a very close approximation. If that's true, can't we just stop sampling early? Say we were going to sample a billion values, but then we stop sampling after the first value. Was that random variable still normally distributed?
Student: I think maybe it isn't.
Teacher: Okay, so at some point its distribution changes?
Student: I'm not sure. I'm a bit confused about it now.
Teacher: Hmmm, well it seems we have something strange going on here. Why don't you have another read of the material on the central limit theorem and see if you can figure out how to resolve that contradiction. Let's talk more about it then.
That is one possible approach, which seeks to reduce the false theorem down to the reductio which says that every IID sequence (with finite variance) must be composed of normal random variables. Either the student will get to this conclusion, and realise something is wrong, or they will defend against this conclusion by saying that the distribution changes as $n$ gets large (or they may handwave a bit, and you might have to lawyer them to a conclusion). Either way, this usually provokes some further thinking that can lead them to re-read the theorem. Here is another approach:
Teacher: Let's look at this another way. Suppose we have an IID sequence of random variables from some other distribution; one that is not a normal distribution. Is that possible? For example, could we have a sequence of random variables representing outcome of coin flip, from the Bernoulli distribution?
Student: Yeah, we can have that.
Teacher: Okay, great. And these are all IID values, so again, they all have the same distribution. So every random variable in that sequence is going to have a distribution that is not a normal distribution, right?
Teacher: In fact, in this case, every value in the sequence will be the outcome of a coin flip, which we set as zero or one. Is that right?
Student: Yeah, as long as we label them that way.
Teacher: Okay, great. So if all the values in the sequence are zeroes or ones,
no matter how many of them we sample, we are always going to get a histogram showing values at zero and one, right?
Teacher: Okay. And do you think if we sample more and more values, we will get closer and closer to the true distribution? Like, if it is a fair coin, does the histogram eventually converge to where the relative frequency bars are the same height?
Student: I guess so. I think it does.
Teacher: I think you're right. In fact, we call that result the "law of large numbers". Anyway, it seems like we have a bit of a problem here doesn't it. If we sample a large number of the values then the central limit theorem says we converge to a normal distribution, but it sounds like the "law of large numbers" says we actually converge to the true distribution, which isn't a normal distribution. In fact, it's a distribution that is just probabilities on the zero value and the one value, which looks nothing like the normal distribution. So which is it?
Student: I think when $n$ is large it looks like a normal distribution.
Teacher: So describe it to me. Let's say we have flipped the coin a billion times. Describe the distribution of the outcomes and explain why that looks like a normal distribution.
Student: I'm not really sure how to do that.
Teacher: Okay. Well, do you agree that if we have a billion coin flips, all those outcomes are zeroes and ones?
Teacher: Okay, so describe what its histogram looks like.
Student: It's just two bars on those values.
Teacher: Okay, so not "bell curve" shaped?
Student: Yeah, I guess not.
Teacher: Hmmm, so perhaps the central limit theorem doesn't say what we thought. Why don't you read the material on the central limit theorem again and see if you can figure out what it says. Let's talk more about it then.