Heh, my 7 year old beat it on easy the other dayThis game is like some sort of abusive relationship. Everything sort of goes fine, then all of a sudden you make one mistake and you DIE HARD SO QUICKLY.
I love it, but good grief. I'm playing on easy and I'm yet to complete it after umpteen attempts.

(albeit I was supervising fairly regularly)
Some tips, in case it helps:
- Use the pause button a lot - treat it like a turn-based game.
- Ditch auto-fire after the first couple of sectors, if not earlier. Time weapons so that they hit the shields at the same time, even if some charge up faster than others.
- take as much time as you can exploring each sector.
- Don't allow yourself to take too much damage, it wastes scrap getting it fixed. Invest in shields/ defence drones/ brute-force quick attacks or whatever suits your ship.
- train your crew. You might get chance to 'spam' the training where you come up against a ship that can't breach your shields. Let them keep hammering away whilst your shield, pilot, engine crew level up. Similarly, if you have a weak gun that can't breach enemy shields - leave it auto-firing.
- upgrading engines is very handy, particularly late game. Against heavy firepower, your shields crumble - dodging 50% helps a lot. I always used to under-utilise this.
- Don't send crew to investigate giant spiders or riots etc unless you get a blue option. It's not worth losing crew to the roll of a dice.
In my opinion, the Kestrel's Burst Laser II is the best bang-for-buck weapon in the game. 3 shots is great vs shields, and only 2 bars of power. My recent Kestrel success involved 2x Burst Laser IIs, a Pike beam, and a small bomb (another favourite weapon)