Government health services are stretched. The health centers that deal with people on a day-to-day basis for minor illnesses are really overloaded and not my idea of attentive medical care.
Dental is the same.
Emergency services are very good, especially for children.
Hamad Hospital, the only government hospital in the country, while well-equipped is badly over-loaded. There frequently aren't enough beds for inpatients.
Getting an appointment with a specialist at HMC can take up to 6 weeks, sometimes more.
Having said all this the government services aren't out to make a profit, unlike the private hospitals and clinics. This means that you don't have to do unnecessary tests, etc for the benefit of these private institutions.
Try to get private health insurance.

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