To the original question: can a run chart show data is normal? Oh, if it was just that easy...we would all have an easier life. No - if it is normal it may appear a particular way, but that is a very high level judgment. Not very convincing.
Here is how it goes.
1. Do a run chart to see if there is any obvious issues to address prior to statistical analysis.
2. Do a curve fit analysis. It is the best way to determine normality.
The attached data shows by curve fit it is not normal.
3. Now the question becomes why is it not normal. Instability? Special cause?
Chart the data - preferably with the correct chart. You stated you took 5 part samples. The attached X-bar R charts show out of control conditions. So, you go back to the copious notes taken during the run (operator change, shift change, any stop/start, any material change, any adjustment) and see if the notes indicate why there was any shifts. If you have reason to remove them, then analyze the data without the shift - but not until you understand the source of the shift.