Why don't they allow people with Coronavirus to use  anti malaria drug, since the president said he has proof that this medication work?

Almost all hospitalized coronavirus patients are getting hydroxychloroquine not only int he U.S. but in most other countries. Trump of course doesn't have proof it works, but that's a different issue. Doctors are using because it MIGHT work and the downside risks are very low.

The president is not a doctor or authority.

Maybe because they know Trump doesn't have a medical degree, did not do research, and is only spouting what his cronies have paid him to say.

The president of the USA does not have proof any anti-malaria drug is effective against the current coronavirus epidemic and is unqualified to assess the evidence even if it had been tested to see if it works. It is currently being assessed in various places around the world. If there is good evidence it works against corona virus, it will be used.

It has side effects include vomiting, headache, changes in vision, muscle weakness and can cause heart problems..  

If the president has "proof", then why doesn't he show it? Why can't he convince doctors?

Because El Presidente is intellectually challenged.