Dead Grass

I planted lawn grass last fall and this spring, and recently there was a 2 week period with unusually hot and dry weather, so about 40% of the last year's grass died, and about 90% of this spring's grass died.  I did a soil test yesterday, and found out that PH is around 7, which is good.  Nitrogen and Potassium are both very low, and Phosphorus is medium.  This tells me that the grass grows very slowly because there is not much nitrogen.  I suspect the grass died because it was not very mature, and the roots were not very deep yet, so it could not withstand the extended dry weather.  I went out and bought fertilizer and top soil.  The fertilizer is the local Country Estate brand by Hewitt's, and the one I am using is the winterizer product with 14-18-14 of NPK.  I first rake the ground to remove the dead grass, then put down 6 bags of top soil on this roughly 100 sq. ft. area, rake it even, apply the fertilizer, apply the grass seeds, rake and mix the soil and seeds, then water until it is very wet.

The recommendation I found online is to make sure the grass get 1 inch of water a week.  Putting down straw will help, but I didn't buy any this time.  The seeding instruction from Hewitt's recommends fertilize the new grass with the high phosphorus fertilizer after 1 month and after 2 months, until the grass is mature and strong.  I am looking forward to seeing the effect of the fertilizer, because this is the first time I am using fertilizer on lawn grass, so I should be able to observe the difference between this planting and last few times which was without fertilizer but with compost.