Hi,
For various health issues, I had to see a doctor in Bangkok 7-8 times (plus 2 more elsewhere) and I always went to a hospital, whether public or private. If needed, each hospital has a pharmacy that dispenses any prescribed medications.
Public hospital: Be patient and bring someone who can act as an interpreter to help navigate the different steps before (creating a file, weighing, blood pressure check, etc.) and after (payment, pharmacy, etc.) the actual consultation.
Private hospital: Shorter lines, and I always had someone from the staff to guide me through each step, even by golf cart if I had to go to different buildings within the hospital.
Many doctors don’t work exclusively in one type of hospital but in both, so you can expect the same quality of care whether you go public or private. "Freelance" doctors also spend part of their consultation time in public hospitals (I think they’re required to do so).