Lightening After Years of Box Dye

You need to start with a colour remover. You can use them up to three times.

Once you've done that you can see what you have to work with. Most likely it'll have some staining from the reds, which could be a problem, but we'll have to wait and see! I think it might be quite delicate if you've dyed over it so often, so you might have to bleach bath rather than straight bleach. This will take longer and you'll have to be prepared to go through some awkward stages, but it's more likely to work than slapping the bleach all over to start with.

If there's too much staining it might be tricky, even impossible to get there, but I find directions turquoise is fairly forgiving, so it might be possible to at least get you started 🙂

I'd start with a colour remover, which is a bit drying to the hair but non-damaging, to see how much of the dye comes out.  Colour removers sometimes work in stages, removing a bit of dye at a time, so you may find more success after two or three times.  Once you've used colour removers about three times, they probably won't remove much more dye after that.

As you've been box-dying, it's best to try to see what is underneath all the dye that is on your hair at the moment.  I don't think you mentioned what your natural colour is, but once dye is removed as much as possible, the peroxide from the box dyes will have probably lightened your natural colour somewhat.

At that point, depending on the condition of your hair, you could start some light bleaching or bleach baths to lighten further if necessary - applying bleach solution over hair that is well-saturated in coconut oil to prevent as much damage as possible.  It's best to mix your own bleach solution with powder bleach and peroxide at either 20- or 30-volume, keeping in mind 30-volume is quite a lot stronger.  Bleach baths are gentler, and it's simply a bleach mixture diluted with shampoo.

To assess the true condition of your hair before touching bleach, shampoo well with a clarifying shampoo to remove any silicone-buildup you might have, then letting your hair air-dry.  Silicones of course make your hair feel healthier than it really may be, and if it is not in a strong enough condition to be able to take a bleaching, you run the risk of frying your hair, which you don't want to do!

EDIT TO ADD:

Probably restated some of what Janine said above, but I took too long to post it... sorry Janine!

Thank you both!

I remember my natural hair color being on the light side of brown, if that matters.

Now to get to reading up on color removers!

It won't matter that much at the moment, apart from your roots. Using permanent dyes will have lightened your natural colour at least a couple of shades. Staining is almost certainly going to be the bigger problem 🙂

Turquoise will definitely be possible, but you may have to settle for a darker or more green toned shade than your ideal. A bright blue toned turquoise requires a pale blonde base, it pulls more towards green/teal on darker bases.
If your hair is stained orange after three colour removers nothing will get it out, lots of people have fried their hair attempting to bleach out stains from permanent dyes. Your other option would be to cut the stained areas off if you are willing to sacrifice some length.
You may be surprised at how well the colour removers work though, especially as your hair is quite short and you didn't colour too frequently.