The strange thing in many cultures is, that it's seen as obscene if female nipples can be seen, even through clothes.
That may be the reason your mother makes you wear the training bra. If it is uncomfortable tell her and ask her to get one that feels better.
(Would like to help more, but as a guy i don't have the knowledge about what bras to use)

Have you asked your mom? Not sure waht type of relationship you have with her, but simply asking her and saying that it is uncomfortable may help out. I've never understood the term "training" bra. Who or what is being trained?

I think the term refers to the fact, that a girl uses a training bra to get used to the feeling of wearing a bra. A little bit strange, because if a girl wears the right bra for her breasts, it feels better with it than without because of the support it provides (if the support is needed). This is not my experience but actual woman with breasts told me that.

I think it's strange that girls of our age wear bra without having anything or nothing at all, just because our friends or our mothers do not look bad in the community, it forces us to wear sport bra or tranning bra. I do not wear bra because I do not need it, it is not comfortable and because my mother lets me choose the height to use or not to use. I think if you do not feel comfortable, you should not use it.

May be is it good to have training bras i think special to learn to wear them.
I’m an mtf transgirl not on hormones yet and not complete out yet.
But in time when i’m expresing my girl side and wearing girls clothing i wear bras filled up with foam cups to have the feeling that i have breasts, and also training my self to wear bras special for the time that im going on female hormones and my breasts start to grow, than ive learned and knowing every thing about bras and wearing bras.
May be should every girl have training bras before puberty starts, so that they can handle brasand learn to put them on and wearing them.

Some women never feel comfortable wearing them. You may not be wearing the right kind/size, either. Some stores have someone who can help you with that.

Sports bras are usually more comfortable because they are made with more cloth and less "prickly" material.

I had training bras but I often didn’t wear them except around family since they wanted me to until I needed a real bra

I just would go in a bathroom and take it off if I had somewhere to put it, and I’d make sure to put it back on before I came home since my mom could tell.