Why do Sikhs wear a turban? Also why are they not allowed to remove it in public?

Thank you for asking this very pertinent question.
This question is not directly covered in Sri Guru Granth Sahib.
Your question is like asking, why do kings wear a crown?
Turban, traditionally has been the dress of the individuals in honored positions.
Sikhs wear it with pride to declare their faith in Sikhi.
Most important is to learn the true Sikhi values and proudly display that you hold these values.
Second part of your question is why are they not allowed to remove their turban in Public.
Once again, it is not that they are not allowed to take off their turban in Public, it is that they chose not to take of their turban in Public. It is in Public that Sikhs declare their faith and values they live by.
Exceptions of Sikhs taking off their turban are when they freed captured Hindu ladies, who were unclothed by the Ahmad Shah Abdali when taking them to Afghanistan. Sikhs took off their turban to clad the unclothed ladies, freed them, and safely returned them to their homes. There are other such example of Sikh taking off their turban and using it as bandage to stop bleeding, using it as a rope for drowning to save someone’s life.