It's easy to understand why Dany gets so much love despite all her mistakes. First, she is still a teenager, so it still plays a major role with the fact that people are lenient with her. She could still learn. You don't expect a 17 years girl ruling like Tywin Lannister or Ned Stark in the North.
Second, the world of Game of throne is so ruthless, sometime with behaviors disgusting, that it is obviously really really nice, and a nice change to have someone powerful caring about other folks like Dany. Seriously, sometime, It's a bit ugh, when all the good men loose and fall one after the other, while those a bit ugh remain... How many nice powerful guys do we have now ? ( No I don't really count Stannis as such )
So yeah, even if Dany isn't perfect, she still has qualities that people appreciate a lot, like caring about her people, and what other powerful men could lack in the eyes of others. Seriously, sometime, I was disgusted when I saw how the common folks were treated like crap.
The world burns because of her, but, hey, she tried to save people, to free people, do you really think that despite the bad consequences, her good actions would be so easily forgotten ? Seriously, when I read in dance with the dragons that Tywin Lannister would have probably burned a whole city with all the innocents and common folks ( children and women and old men too ) inside just to find only one guy, Robert, that made me like him less than I used to. Of course, even if it is really ruthless, I can respect this guy, but if there is someone different like Dany, who would never dare doing such thing to innocent people just for one guy, yep I am going to like her.
Her idealism, even if some can find it absolutely stupid can be shared and respected by many folks. I recall that this is this kind of idealism in real life that broke many disgusting systems in the past, sometime you need blood. It is inevitable.