Reading the above it's great to see there are lots of possible ways to get the job done. I just scrapped through my first normal victory after a good few attempts and a slice of luck. Would agree with those above saying that boarding parties are the easier (not easy!) way forward both before the boss fight (getting extra scrap) then during (you can take out weapons etc). I also found the cloak invaluable in avoiding damage. I think using a boarding party and cloak might be easy(ier) mode for normal
I didn't have great weapons or any drones and was only able to field either a Hermes launcher/Hull Laser II or deactivating one of those and running a Healing Burst/Fire Bomb alongside. The fire bomb proved very effective on the drone stage which was a good job given the launcher was rendered completely useless until the drone control was down.
The last fight went down to the wire even though it started with no crew (should have left 1 alive so no AI repairs???). I went in heavilly damaged with only some strategic cloaking getting me through to the death where I was pounding away the shields with the launcher and the hull with the laser. I feared the worst as its' launcher came online with me down to 3 hull and the cloak on cooldown but a large slice of 3 miss luck later and the hull laser finished the job.
Another thought that occurs is to consider going with the Osprey as that may make it easier for some. You can focus on defence, while the artillery cannon does guarenteed damge every shot, no matter what shields remain.