Nitrogen is not the only nutrient needed to good garlic growth. A good example of a balanced fertilizer for onions and garlic (side dress the garlic) is from Dixondale Farms as a 10-20-10 mix of NPK. Straw doesn't have enough of any nutrient to help garlic grow better on its own., and just adding nitrogen will give you something other than a healthy garlic.

I've found that straw isn't one of the worst offenders when it comes to tying up nitrogen, until it gets incorporated into the soil, anyways. It is one of the best mulches for attracting healthy earthworm populations because it provides them with food and protection 24/7.

If you are buying commercial fertilizer, you are already paying(via a tax levied on all fertilizers) for a free soil test available from your county extension agent. Get your soil measured and you'll know where you stand.

Overfertilization by commercial nitrogen will generally manifest itself by a "burned" appearance, especially on leaf tips.