I am currently a year-one history PhD student at a UK University. I didn't get any fund from the school or outside. I just supported my study by myself relying on my personal savings and some helps from my parents.But the tuitions fees for international students in the UK are very high. The school cannot provide enough financial support for my outside research in China, the US and some other places. I feel difficult to continue my PhD now and I want to find a new institution which can provide me with a fund. But I have no ideas about how to talk with my current supervisor who is so kind as to help me to improve my research proposal for year one. Also, I am not sure the supervisor's attitude to the student like me in the new school. Can you give some advice for this? Thanks.
You seem to be in a box that is difficult to escape. You need funding and it sounds like you can't continue without it. So, think about that first, not about changing places.
I think the way to approach it in the short term is to sit down with your advisor (i.e. face to face) and say just that: I need funding or can't continue here.
After the conversation starts let it flow. Give the options you think you have, but ask if there are other options. The advisor probably has more experience with this.
Perhaps the advisor can either find you that funding, or suggest a way, or suggest a better place to continue your education. They can be a resource in figuring out what to do. Perhaps there are TA/RA positions open. Perhaps there are grants that apply.
If that fails, then look for another place, but make sure you have adequate funding before you jump.
I sympathize, but why did you start this PhD without funding in the first place?
Imho the problem is not about staying or leaving, it's about having a real funding plan in any case: Finding funding for a PhD in history is not easy, but in order to do a PhD one needs at least 4 years during which one has to pay living costs and university fees, possibly also research costs such as travel and conferences fees. These expenses are fairly predictable and it wouldn't be very wise to keep burning your savings (or your parents' savings) without any guarantee that there's enough to reach the end of the PhD. In other words: be careful about the sunk cost effect.