There are quiet ways to practice, so you can still improve without letting other people hear you! It's not always about your voice that needs to be trained. If you're still too shy to sing, you can practice breath control since it's pretty quiet, stick your tongue out to train to relieve tongue tension, or just do what I do sometimes when there are other people in my house and I want to practice singing. Put your hand on your mouth and do some exercises if you know any (but then again... you'd probably have to learn what the exercises would feel like first, and the only way to do that is if you can hear it properly...). And then maybe one day you can take your hand off your mouth and just simply start and experiment with your voice. Try humming. I remember it was much easier to be on pitch when I hum than when I sing, so you could be less scared when training your voice.
The main thing in getting over anything that makes you shy in general, is to expose yourself in that thing (so your brain can learn that it's safe to do this thing, and it's nothing you should be afraid of). So the only way to get over being shy to sing, is to sing. It's not always going to be the same for everyone on how fast they build confidence. But just do whatever's going to build your confidence. Like, if we're going to use Mariah Carey's "Touch My Body" as an example, first you can just sing one word "touch". And then try singing two words "Touch my", then three words, then four words. Then finally, you can sing "Touch my body, GIVE ME WHAT I DESERVE!" in your room, and your family will think you're a weirdo. This dog in my profile pic looks really cute.