"Maybe the prime minister could have explained this at the hospital."
Translation:Başbakan belki bunu hastanede açıklayabilirdi.
I'd say it is the word, it should be anlatabilirdi in your sentence. It is constructed as:
anlat (verb root) + (y)a (possibility suffix aka 'can') + bilir (from bilmek) + di (past tense)
The word order is fine, though the stress of your sentence is different. Keep in mind though that anlatmak actually means to tell/to narrate something however it is often colloquially used like açıklamak/explain.
Hope that helps :)