There have been a lot of complaints on the Pirate 101 forums that it doesn't feel as though we level fast enough and that the quests don't give enough experience points. The answer from the developers is that most of the xp you get is from the main story quest line and they aren't lying. You will never level faster than when you just focus on the main quest. However it can still feel like you aren't making any progress.
An example of this was, while playing last night, I had just completed every quest line I could find in the Skull Island Skyway and was finally set to move on to the next world of Monquista. I looked in the quest log at the two quests I had left to do. One was a side quest to go to the capitol of Monquista and see about a religious figure and rewarded 10xp. The other was the main story quest to go to the same place and rewarded 350xp. I look down at my xp bar and see that, being lvl 13, I am around 200xp away from getting lvl 14. I think, "great. I'll go turn in those two quests and take a break when I level up." An hour later, I still hadn't received that 350xp. This is because the game tells you what you will receive when you complete the quest line, but there may be 30 checkpoints and requirements between where you are now and when you get that sweet, sweet experience dump. In this case, it said to go to the capitol and see the king about your prisoner, then, after going through that, they send you to a prison island to deliver the prisoner. Next you have to free the prisoner and fight the guards. Then you have to go to a third location and check in with the resistance. Then you have to go see the deposed queen. It is only when you have spoken with the deposed queen that you see any quest experience at all.
Here's my suggestion: split that xp up over a few more points. You don't have to get xp at every single stop on the way, but who would object to 100xp for speaking to the king, 100xp for freeing the prisoner, and 150xp for talking with the deposed queen? That way, you feel as though you are making progress, though the end result is the same. It's curious that they designed it the way they did, in a way where most quest turn in's don't result in xp gain since they are just steps towards an end goal. Since most of what keeps players playing an MMO is psychological, you would think they would turn the methadone of xp to a steady slow drip instead of sporadic bursts after long dry spells.