There are a couple of answers to this. Firstly, your back muscles will need development before they are happy with the lower position. Time and training will allow this. You may not have this priviledge in the time available. I find that when my back starts to tweak, stretching the back while on the bike helps as does forcing my heel down while in the saddle - effectively overextending the leg.
But, you may be able to raise the handlebars. Firstly, the stem (the straight bit between the forks and the bars) can be inverted to provide a small increase. They will work equally well upside down.
In addition, there are often spacers on your steerer tube. These can be placed above or below the stem. Moving the spacers from above the stem to below it will increase the height of your stem.
However, take care when tightening up your stem and top bolt. When tightening up the bolts after moving the stem, you should tighten the top bolt (the one that goes down the middle of the steerer) before the stem bolts. In addition, be careful that you do not overtighten it as you this affects the steering of the bike.