I am trying to find just one example which can satisfy the above, so far I have run into dead-ends! The biological data tends to be skewed to either side. Any guidance would be highly appreciated!
You can't prove that you have a normal distribution by looking at data; there will always be some other, more complicated distribution (an infinite number of them in fact) that are closer to the distribution of some sample than the sample is to a normal distribution.

You can, with enough data, show that you don't have normality. Indeed often you can tell for certain with no data at all (most biological measurements are strictly positive)

As with any distribution we have named, normality is a simple model, an approximation, not a biological fact.
If you are asking about 'exact' normal distributions, there are as many examples as examples of circles or triangles in nature.
It helps to remember some properties of normal distribution.

- It’s continuous.
- It can take any value from negative to positive infinity.
- it’s perfectly symmetrical.
- It has (excess) kurtosis of 0.

Now, do you think there is a real life phenomenon, that can satisfy all of the above?
IQ?

Is that cheating because I think it is a normal distribution by definition...
True.
Height of a population
Okay let’s say you run an experiment measuring some continuous bioindicator for fish that are exposed to some drug with the control being fish not exposed to the drug.

If you fit a linear regression and you pass the checks for the assumptions of a normal dist (normality conditional on independent vars and constant variance via Shapiro test and Bartlett test) , then we can use the normal distribution to approximate the situation and draw conclusions.

We usually check if things are normal because it’s easier to defend your results when you are using a well proven method compared to some beta regression or other weird glm.
When I taught intro stats I’d often use cholesterol as data that are close to normal.  There’s some nice histogram examples online.
Well, if you assume hat the values of any biological phenomenon are bounded below and above, which might be true (?), then no biological phenomenon is normally distributed in a strict mathematical sense.
How about the publicly available iris dataset using sepal length etc... Surely these are close approximations to a nie.al distribution. You can see qq plots and run a Shapiro wilks test to verify.