I have no doubt that God does intervene on occasion to heal people without the aid of medicine -- but nowhere does the Bible urge us to reject medical treatment. God has given us the ability to develop modern medical treatments, and we should see them as a gift from His hand.
It's interesting that Luke, who wrote two major books in the New Testament (Luke and Acts), was a physician; Paul called him "our dear friend Luke, the doctor" (Colossians 4:14). Luke witnessed a number of miracles while traveling with the Apostle Paul, and he realized that they came from God's hand. But he never rejected the use of his medical skills, because he knew God could use them to bring healing.
This doesn't mean, however, that we shouldn't pray when someone is sick and ask God to heal them. Ultimately, our lives are in God's hands, and even when He uses medicine to bring about a person's healing, He still should get the credit. The Bible is clear: "From the Sovereign Lord comes escape from death" (Psalm 68:20).
You are right, however, when you say that God wants us to have a strong faith. But how does our faith grow? It begins with our personal commitment to Jesus Christ and His will for our lives. It continues as we grow closer to Him through reading His Word, the Bible, and through prayer and fellowship with other believers. Is your faith in Christ growing stronger every day?