Cyre is right. You just have to tactically outmaneuver those ships.
I have already lost one ship too the "Cursed".
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Easiest way to get it is to team up with a crew (yeah I know always a good idea) and level up your sailing skills a bit, a big ship (Std or War class Galleon or Frigate) helps a lot too.
Depending on the quest (I think I've had to get like 30 cursed woods or battle touched water etc. in all those quests that are hidden everywhere...) you can team up with a higher level player on their warship and if you get at least one cannon shot in you're eligible for quest items when it sinks so if you find someone who is any good at sailing and has a big ship just stow away is the easiest and safest option. If you see me on just ask for help
Now if you are dead set on going it alone there are a few things that will make life a bit easier.
Go after the lower level ships, Phantoms and such and avoid the higher level ships.
Max out your cannons and add at least one point to turning and wind catching (the passivize sailing skills) and also a point to the open fire command as soon as it's available, it really deals out the damage.
If you can find a ship that's already damaged, or engaged in battle help your fellow pirate out and fire on it. He may not appreciate it much but after all your a PIRATE steal some exp (and hopefully cursed wood) in the process.
Use your turning skills (and smaller ship) to outmaneuver the larger vessels, stick to the stern if you can you can do more damage that way and he cannot shoot you but keep a safe distance!
If you see them firing the special ammo stomp on the breaks and you can usually avoid most of it, if not all of it this way just be sure to get moving again as soon as possible or they will open up on you again.
It's tricky but if you can use the left and right arrow keys only when you are circling a vessel and keep the mouse camera on it at all times, quite often almost every volley the NPC fires will miraculously miss you if you do this... No idea why it works but it seems to work every time for me, after discovering this trick maxing out my sailing skill was a piece of cake.