"I'm getting better." or "I am getting better now." would be much more acceptable. "I get better now." doesn't indicate that your improvement is 'in progress' (as the other examples show) and it sounds somewhat stiff, stilted and unnatural.

'I get better now' is NOT a correct or appropriate answer to 'How do you feel?'. A proficient English speaker would never say this, and if you chose this option in an exam it would be marked wrong. In the phrase 'get better', the verb 'get' is a dynamic verb. This means that it has to be in the present continuous when you are talking about what is happening at the present time. The correct response is 'I'm getting better'. 'How are you?' 'I'm getting better.' Remember this and use it.