OP, if you want this sorted so that the behaviour stops, then please do not follow the adversarial approach suggested in some posts. Do not ignore this so that the behaviour escalates.
Just make quiet enquiries first to establish what the company's policy is. If it has one, then the only time you could be treated negatively is if you are making a malicious accusation. That is not the same as reporting what you feel is inappropriate behaviour and an employer feeling it isn't bullying after an investigation.
Too often people either sit on these things so that they escalate or they immediately opt for the formal approach which can cause more harm than good.
The assumption seems to be that the supervisor is bullying. I'm not defending them at all or saying you are lying. I'm saying that sometimes the context and content of things can be lost, implied, taken wrong, etc.
An informal route will give you the option to present why you were upset by the comments and emails and why you regard them as inappropriate. It will also give the supervisor the chance to perhaps see how their manner can come across and change their behaviour. Nothing adversarial and more common sense and common courtesy.
If after this you aren't happy with the outcome or there is no change, then you can look at making it more formal if you wish.
The important thing is to get the behaviour stopped, not become involved in a battle.
Last and to emphasise yet again, you are protected under legislation so that if you make a complaint you cannot in anyway be penalised.