Hi, JanieV!
Often times funky or more unconventional colors are available in semi- or demi-permanent formulas (although there are some brands now that are startng to make permanent formulas, these tend to be available direct to salon/licensed professionals versus being available to the mass public in general retailers or even places like Sally's Beauty Supply). These formulas can last anywhere from 4-12 washes, so depending on how often you're washing your hair, color may fade faster.
It's also not out of the ordinary for the porosity of hair to have been affected by the initial bleaching process. This can affect the hair's ability to then hold onto and retain the color element that is put back into them after the bleaching step. If hair is too porous, color can essentially just "seep out" quite easily as the integrity of the hair won't allow for it to "hold" or 'stick".
In your case, for dark hair, depending on the color you're wanting to go, bleaching may have to be required. For example, if your hair is black or even a deep chocolate color and you're wanting pastel pink, turquoise, emerald green, or even flame red added in, bleaching will be needed to lift the original pigment of your hair so that the colored dye can deposit and thus show up. Depending on the type of dye that is used, if it doesn't require any developer, then that means the dye formula itself doesn't provide any "lift" to the hair's color (brands like Manic Panic or Beyond the Zone are examples of this) so these formula only focus on depositing and incorporating color. If you're color product requires you to utilize developer, then there is a certain level of lift that can take place depending on the volume of developer used. The higher the volume, the more peroxide or lifting component is used to help the color penetrate into the hair shaft. For example, if someone with naturally medium brown hair is looking to go to a milk chocolate color, they may purchase a milk chocolate color and use a 20 or 30 volume developer to help lighten their natural color slightly and also help that shade deposit. Since that color change isn't very dramatic, no bleach is necessary and the lifting abilities of the developer should take care of providing that difference in the hair color look (you normally don't want to go more than 3 to 4 shades lighter without the aid of bleach as you may run the chances of unflattering undertones appearing and affecting the actual color you're trying to achieve, so in other words if your hair is black but you want to go to a sandy ash color, doing just hair dye with developer won't cut it).
It's also advisable that after a color process (be it with semi/demi-permanent or permanent formulas) that you don't wash your hair until 3-4 days after so that you don't run the risk of prematurely washing out trace aspects of the color and cause it to begin fading. Aftercare products like using color-preserving shampoos and conditioners, as well as deep nourishing masks are also crucial in helping to restore the integrity and strength to your hair as chemical processes can be quite taxing on strands. Avoid super hot water to prevent drying out strands and using thermal protectants when using heat tools are also practices that can help prevent additional stress or damage.
For the various services that you have experienced thus far, after the funky color had washed out, do you feel like the portions that are bleached are indeed light enough to where the color you want to add can show up to the degree you want? In other words if you're wanting teal added to the bleached portions, where those portions a very pale ashen blonde or even a platinum blonde to where they soaked up the teal and it was super loud or were the bleached portions very yellow, brassy, or even coppery in color to where when the teal was applied it came out more dingy and dusky looking? If the bleached portions aren't light enough to have the funky color applied on top show true to what you're wanting, then unfortunately, additional bleaching procedures will need to be done; however, if the bleached regions are light enough then you may want to merely look into some semi/demi-permanent color formulas that may have some more holding power than what was used in the salon (which, by the way did the salon disclose what brand of color they used?).
How long ago were the bleaching/coloring processes? It's important to not do too much too close to one another as that does tend to stack on the damage, so if you do need more chemical processing, I'd recommend booking a consultation with a reputable and licensed hair dresser to first get an evaluation on your hair's current status (disclose everything from what was done to how long ago to any hair care practices you have) and then determine how to proceed.