There are 3 words in the New Testament that are all translated into English as "heal." They're not all the same things:
The first one describes healing generically (iaomai)
The second one describes a miraculous "faith healing" (sozo)
The third one describes treatment for a sickness as a course of therapy (therapeuo)
And now... the story of Jesus healing the centurion's servant makes more sense:
Matthew 8:5: When Jesus had entered into Capernaum, a centurion came to him, beseeching him, 6 and saying, Lord, my servant lieth at home sick of the palsy, grievously tormented.
7 And Jesus saith unto him, I will come and therapeuo-heal him.
8 The centurion answered and said, Lord, I am not worthy that thou shouldest come under my roof: but speak the word only, and my servant shall be iaomai-healed.
10 When Jesus heard it, he marvelled, and said to them that followed, Verily I say unto you, I have not found so great faith, no, not in Israel. 13 And Jesus said unto the centurion, Go thy way; and as thou hast believed, so be it done unto thee. And his servant was iaomai-healed in the selfsame hour.
Jesus offered to therapeuo-heal him - that is, to come to his house and oversee a course of treatment nursing him slowly back to health. The centurion says that Jesus should simply command it, and he will be well. And with that kind of faith, that's what happened.
Also, what was Jesus profession? I don't think He was a carpenter... the people didn't come out in droves to get wagon-tongues from Him. They went to Him for healing... He was a doctor.