When fighting against a tug-of-war (between scientific and "alternative" medicine, say) in a high-dimensional space, the best strategy is to pull sideways.
Fortunately, homeopathy is mostly harmless—a wonderful improvement over the usually actively-harmful "medicine" of its day, which is why it gained enough of a following to kind of get grandfathered in to our current system. So you probably have some time. If you're not a minor, you have the absolute right to refuse your mother's "treatments". It sounds like it's for your own good in her mind, so she won't disown you over this. If you are a minor, it's probably not going to kill you, but you can still fight it and maybe consider early emancipation.
But the level of insanity required to believe in this stuff is also sufficient to persuade people to drink bleach and die, or donate all of their money to Nigerian "princes"—a very dangerous place to be in the Internet age. Homeopathy is only a symptom. The real disease is faulty epistemology. It's endemic, but some people have a worse case than others.
Irrational people develop defenses against direct arguments, but you can often work around these using the Socratic method.
Street Epistemology is a modern method of Socratic dialogue focused on improving the interlocutor's epistemology, guiding them to eventually doubt and release their attachment to false beliefs. For someone in your position, it's worth learning to do. You may want to practice with others before trying it on your mother though.
You might propose a compromise. You don't get any injections but are willing to take sugar pills in high enough concentration so that they are practically inert.
Afterwards, the treatments won't have any effect on you, so it's okay to take them to allow your mother to express her care for you.
Don't attach your identity to beliefs about things that don't really matter.
The problem your trying to solve is not how to change your mother's beliefs. Your problem is how to communicate that she's making you feel negatively and if the two of you are going to have a relationship she needs to change her behaviour.
Trying to have a system 2 scientific discussion with your mother in this scenario is playing water polo with a lead ball. You may go in with a clear head and a scientific argument and manage to throw the ball. But 3 sentences in and you're both going to be below water, having an emotional system 1 conversation.
What is an emotional conversation? It's a conversation in which:
The goals of the participants are not actually to present facts or reasoning about the world (e.g "homeopathy does not do the things you claim and here's why"), but to communicate how the actions of another makes them feel ("when I say that homeopathy is not for me but you push it on me it makes me feel disrespected and frustrated")
The skills needed to facilitate the conversation are less reasoning and language skills, and more being able to express how your feeling in realtime, communicate you boundaries, being sensitive to how the other person is feeling without taking responsibility for it, and be vulnerable if that's appropriate.
This is all easier said than done, especially when it comes to family. Some things you can do to practice a conversation like this:
Have it in your head, and imagine your mother saying whatever would maximally trigger you and how you might respond with emotion communication and boundry setting. ("I feel disrespected" -> "it's not my fault you won't see the light of homeopathy" -> "now I'm feeling like you don't care that you're making me feel this way")
Role play with a therapist or trusted friend. Some universities or work places offer free therapists that could help you if you're insurance won't cover it.
Read books on the subject like "non violent communication"
Good luck my friend!
Honestly, I think your chances to change your mother's opinions on homeopathy are close to zero.
(Unless perhaps it is that case that someone else is pressuring her into homeopathy, and you could somehow undermine her trust in that third party. But that is manipulation.)
I would recommend setting and protecting your boundaries. You should have a doctor, who is different from your mother's doctor. Then, refuse to discuss medicine with your mother. Do not try to convince her, just refuse to take any pills or injections. Not "because they don't work from the scientific perspective", but simply "no". If she insists that you are stubborn or stupid, let her have the moral victory, but keep saying "no" to the medicine.
The way I see it, I can only grit my teeth and hope for enough income to move out as soon as possible.
Moving away from your parents can change things a lot. I still remember the first morning I woke up at my own place. It was maybe 10 AM, sun shining through the windows, a complete silence, no one was yelling at me or at someone else in the next room, and I thought "all the following mornings will be like this" and I felt happy.
Yeah, it can cost a lot of money...
Explain to your family that to properly evaluate homeopathy your entire family needs to understand statistics. Give your parents some books on statistics and whenever they bug you to do something about homeopathy ask how much progress they have made with the stats books. Start numerous conversations with your parents about statistics and work statistics into all the conversations they have with you. When they want to watch TV suggest instead they watch a YouTube stats video. Eventually, work out a truce.
Changing someone's mind is an incredibly difficult thing to do. In this situation, I would ask myself: is this what I want to pour my time and effort into? Is this the most important thing for me to do right now?
What has worked for me in the past is re-evaluating the situation. Assuming that neither of you will budge, is there an equilibrium you can reach--both now and when you move away? "Walking away" often feels like the only viable option, but in reality, there is usually a whole spectrum of possible options. What would work for you? What are the topics you care and that you want to share with your mother? What are the topics she cares about and wants to share with you? (Is there overlap?) The answers might lead you to finding a new equilibrium, even if it's "meeting my mother twice a year, talking only about food and pets."
Also, because I read the last sentence of your post in a somewhat struggling tone (which may not be true, because words on a screen), I can share that doing a few sessions of talk therapy can be really helpful in a situation like this.
The Prediction-based-Medicine paradigm would be another way. You ask for predictions of what happens when you take the treatment and for predictions about what happens when you don't. Then you might flip a coin to decide whether to take the treatment.
This approach can allow you to see how the Briers score for the treatments compares to the Briers score for not having the treatment.
I find the concept regression toward the mean very useful because it is very intuitive, understandable and not related to health. Especially this example of Kahneman about an instructor of the Israeli Air Force is totally relatable, funny and understandable.
The fact that this concept (and the example) is not related to health has the advantages that one doesn't need to have any kind of health-related knowledge to understand it and that it may not trigger usual arguments. The disadvantage is that they may argue that it does not apply to health. But there is a workaround: start by pointing out that sometimes one does feel ill and this feeling goes naturally away --the body is a healing machine! They will have to agree on that. Then you can tell the story.
I tried something similar a couple of times online with total strangers... I'm really not sure of the outcome. But face to face it may work better, especially with family members.
Good luck!
My mother is a believer in homeopathy, and at the present time I am living with my parents.
I am frequently pressured to take homeopathy, usually in the form of injections.
To cure cold, to boost immunity (a very hot topic in the current pandemic), etc.
My efforts to talk science have been futile, most of the time I am offered to do it and see for myself. While I like testing things empirically, I much suspect she won't be able to update her beliefs after 20 years of "successfully" treating herself and others. All papers I cite are "paid for" and "influenced". Well, you most probably know all the arguments already.
This is getting on my nerves, plus she's often rude about my rejections. The way I see it, I can only grit my teeth and hope for enough income to move out as soon as possible. However, it's rather difficult to walk away from your family (though I suppose I won't last much longer).
Is there a better way?