PhD application: bad grade in relevant subject at master's

I believe it is very normal to have some drawbacks in every university experience, especially in the situation you described. I believe most people would be understanding if you explained to them what caused that specific grade (the changing masters and countries, etc), and I would explain how you are improving (extra hours of study, asking for advise of older students, learning better organization techniques, things of that kind). It's all a matter of how you expose the situation; if you expose it with a positive output, they will get a positive impression of you!

For my master thesis internship interview, one of the topics I talked about was how I in one semester had very low grades (average of 13/20) and even failed at one subject, and how I decided to turn things back around, got up on my feet and in the next semester, did one extra subject and ended with an average of 18/20, and they loved it, I was actually their first option for the internship! It's all about how you deliver the info :)

I can relate to your situation, as Johanna stated above, it is normal to feel like you might not be on par with other applicants. As far as lower than desired GPA, or any other factor that you may be holding you back, one of the things that the Ivy Leagues (and other schools in the US for that matter) generally require for PhD admissions is a statement of purpose, where you explain why you are applying to the programs you are applying to as well as your research interests. It's been discussed in a couple other posts on this site, but the statement of purpose (SOP) is a great place to explain any hiccups that may have occurred during your previous education (people tend to be more understanding of stumbles along the way than you may think). So in the case of your low grades, you could briefly mention your low grades, and what caused this and how you improved from that situation and how you plan to build on this during your PhD studies. You want to remember, though, to not bring too much (negative) attention to this and keep everything in a positive light here.

You are far from the only one I've seen on here to post worrying about lower than desired grades; a lot of people have posted fretting not being able to get into a PhD/graduate program due to lower than desired grades, most often owing to a bad first semester or even first couple of years. However, a lot of people have overcome such situations and have been able to secure admission to great graduate programs, so keep your chin up and eventually things will work out. Hope this helped and best of luck.

This is exactly the sort of situation that a reference letter should address. Find a referee, perhaps the lecturer for that subject, who can credibly say that your results don't truly reflect your ability and explain why.

(Of course, as other posters have suggested, you can also explain the situation in your own statement. But this might look a bit defensive, which is why it's so helpful to have someone else, who can take a more objective tone, do it for you.)