If you want to get some early damage on the Royal Fortune then immediately after going to sail use the Mortar. It is also recommended to mark it with the Spyglass so you can see the distance.
Do not directly follow the Royal Fortune. The passage it takes between the rocks will collapse just after it goes through. The Jackdaw is forced to go around to the right. Go to full sail and turn to the left after you clear the Sandbar. Turn West and go full sail again. Get through the sand and rocks and you will be next to the Royal Fortune before long.
At this point the Fortune will start to drop Fire Barrels in a line behind it. Keep to its side as long as you can. After that, start using the Swivel guns to explode them before they can damage the Jackdaw. You can alternate between Chain and Mortar fire with some careful speed management to maximize the damage you are doing.
When the ships make it out of the channel they've been in, things go from bad to worse. You will find a number of Spanish ships that have come hunting the Royal Fortune. Only bother with the other ships if they pose a direct threat to the Jackdaw. otherwise just keep your fire on the Fortune. When you do disable it, move in quickly and board it.