If granular fertilizer, you should water in the fertilizer so it can get to the roots, or apply right before a rain.
If you can, mulch the clippings and leave them on the lawn, those are packed with nutrients for the soil! (just not clumpy)
Fertilizing after you mow is important because you want the product typically to be able to reach the soil line. If you fertilize on top of the tall grass the product can be wasted and in some cases even burn the grass leaves.
Some other things to note would be that you should never cut more than 1/3 of the blade of grass off when mowing, and be sure to always water your lawn if it’s been a while without rain!
You need 1″ of water per week naturally or by you!
Use a sprinkler
Use a container to test how much water has fallen from the sprinkler
No matter how much you fertilizer your lawn or how perfect you mow it, if it doesn’t get enough water it will go dormant or die. It is important to get at least 1″ of water per week when you consider natural rain and watering yourself. It is also more important to water completely that frequently. If possible, water for ~20 minutes so the water can soak deep into the soil driving the roots down vs. a light watering teasing the roots towards the surface water. This is why using a sprinkler is the most effective way to water a lawn. It can be time consuming to stand there hand watering, so setup your smart phone alarm for 20 minutes and go do something else.