Stuck on Dragon Rising quest

The briar hearts in this quest are apparently prone to glitches. I found this thread where people are discussing the various ways in which they were able to break this quest. In some cases, when they got notification that they needed to get that specific briar heart, all the briar hearts they already had became quest items.

I think the best you can do is load an earlier save, or hang on to this quest until Bethesda patches it to work properly. Even so, you might not be able to finish it unless they create some way to reset your progress in it or otherwise fix the quest if it is already in progress.

You are just too strong for some of them. I began following the main quest line as soon as I started the game. My difficulty level is either Normal or Hard, I forget which. I fought my first dragons at level 4 ~ 6, maybe even earlier. My damage at around that time was less than 20. I've come across random ones while on my way to complete various quest that were not that strong. Me and my follower took them out very easily.

So it is no surprise that you can one shot a dragon, as it could be one that is meant for a level 6 to fight.

Also note that brown and green dragons are weaker dragons compared to their red and black.

That, or there is a bug in the game.