It will definitely get better! Over the next few weeks, your supply will start becoming more regulated and your baby will learn how to handle it. In the meantime...
You've got a lot of good advice below, especially about making sure that gravity isn't making it worse. I nursed my baby with him almost verticle (as opposed to the horizonal cradle position). I would have him kind-of straddle my thigh, his head propped against my bicep, my hand behind his back with me reclining just a little. This was better for him than the cradle position.
What worked best though, was pumping a little before feeding him. I got an inexpensive hand pump and used that for a minute or so before feeding him. It took care of the initial forceful letdown, helped with the foremilk/hindmilk imbalance issue because I was pumping off some of the foremilk, and also softened me up so that it was easier for him to get a good latch (I also had an over supply issue, and often became engorged making latching more difficult for him). I would count, and if I did 50 'squeezes' with the pump, I could get as much as 2 ounces of milk! There was still plenty in there to feed the baby on just one side. And I froze down all that extra milk to use later (great if you are going back to work, and even if you aren't you can use it later when you start making baby cereal, etc).
Once my baby was older and better at latching and my supply regulated so that I wasn't engorged anymore (probably around 12 weeks), I stopped pumping before feeding. Like your baby, my baby has always eaten very fast - 10 minutes tops. That isn't necessarily a problem. Don't worry about making her eat for longer. She'll eat until she's full and that's all that matters, not the amount of time she's nursing.
If you have any more questions about this, feel free to message me.