It does not seem normal to me to just let things go and not worry about them. But, as you go on through life you realize you can't do everything. This is something I'm beginning to see as I go through college and things are pulling me in every different direction and I really have no clue what is best.
I have started to pray more and more as I have drawn close to God on my own, not because my family is making me or my parents are making me go to church. And this comes with learning things on my own, like what is God really expecting out of us and what does he mean when he tells us to trust in him.
Basically, what I'm getting to is you can't pray and be worried. They don't work together. It doesn't make sense to ask Him for something and then behind his back keep questioning if he's going to get it done. I have experienced the great things God has done in my life when I let him do it freely. Having faith is a major part of allowing God to do what needs to be done. Faith is almost equivalent to trust. Like, when you're a kid and you're swinging and you keep counting to 3 and you never jump... you keep swinging. You keep saying 3 and you get comfortable at the height you're at but you need to take a leap of faith and God is not going to let you fall. There are decisions that need to be made in your life, and there is more danger in swinging than there is in jumping. So, when you want to worry, pray instead. But, know that prayer and worrying do not work at the same time.