I've yet to see any sort of evidence for miraculous healing. It's also a strange coincidence that god can't heal things that medicine can't either.
Why can't he cure AIDS or Down Syndrome? Or even grow back an amputated limb?
He only seems to have the same healing powers as a Doctor.
Firstly, there are different kinds of 'illnesses' in our world. There are physical illness, mental illness and spiritual illness.
There's also different causes, such as generational iniquity, demonic influence, trauma, accidents, sin.
You mentioned AIDS, this is can be due to physical blood transfusions, or from sexual sin. For all sin has a consequence. Unfortunately sometimes it's the person who has been sinned against that suffers (eg rape victims).
You mentioned Downs Syndrome. This is a genetic, physical thing. It's not a curse of God. I know a guy who was born with the umbilical cord round his neck. The lack of oxygen caused damage to his brain, and though he is physically 29, he is mentally about 5. We don't live in a perfect world. The thing is, that when he gets to Heaven, he will have a new body. Something to look forward to.
Like I mentioned before, rape victims suffer as a consequence of some one else's sin. It's not just a physical thing, but emotional and spiritual. The body heals a lot faster than the spirit and emotions. And (with personal experience) it's a lot of forgiveness, and allowing themself time to grieve and express the emotions involved. God does heal.
It's like the story in the Bible about the lame man. Jesus said to him, 'your sins are forgiven - get up and walk' and he did! Sin is crippling.
Generational sins are real - have you ever met someone who's grandfather was an alcoholic, who's father was an alcoholic, and they were struggling with alcohol as well. In Exodus, it mentions that the sins of the fathers would be passed to the 3rd and 4th generation. This is true for things like witch craft and the occult. The thing is, that we need to forgive our ancestors for the things that they did wrong, and ask God to sever any un-godly ties that run down the generational line - asking that anything that has come through those lines would be taken, and anything stolen from self, to be returned.
It's something that the church (on a whole) doesn't pay a lot of attention to. And those who do, are over-the-top, to the point that they are weird/annoying.
Jesus spent 3 years teaching his disciples. He showed them His power, he taught them to use that power and authority, then told them to go forth, healing the sick and casting out demons. What would've been the point, if it wasn't needed any more? What would be the point if once Jesus died and rose, that everything was fine.
Also, the Bible says to put on the Armour of God (breast plate of righteousness, belt of truth, footgear of readiness from the gospel of peace, helmet of salvation, the shield of faith and the sword of the spirit) for the enemy is not of the flesh, but of the spiritual realm.
We live in physical bodies, which can be injured, and sick. But the real sickness is that of the spirit.