Straightening damages your hair and damaged hair breaks, so even if it's growing it won't look like it is because all of your ends are snapping off. Take this from a girl who was straightening her hair and trying to grow it for around ten years lol. I never got past shoulder/arm pit length.

Now that I'm natural my hair grows at the speed of light ... well, okay, slight exaggeration, but it grows A LOT quicker. I recently cut off all my straight ends and my hair was earlobe length and now it's shoulder length and it's only a month later. Point being, healthy hair grows and damaged hair just snaps, so you need to start nurturing your hair.

Some tips:
Use oils like coconut oil that really penetrate your hair and moisturise.
Cut out the sulfates.
Deep condition once per week.
Cut out heat or use it A LOT less.

You should try just doing two pigtail plaits in your hair overnight for nice, loose and defined waves. Or other heatless styles for curls. That's what I did while transitioning. It'll stretch your curls to waves, which is what you said you were aiming for and help you to cut the heat

Curly type: 3A Botticelli curls. Medium porosity / density.
Pre-poo: Organic Virgin Coconut Oil
Low-poo: Shea Moisture Coconut & Hibiscus Curl & Shine
Conditioner: Faith in Nature Raspberry and Cranberry
D/C: Shea Moisture Raw Shea Butter Restorative, mixed with honey & other goodies
Leave-in: My DNA
Styler: Umberto Giannini Scrunching Jelly / Ouidad Climate Control Heat & Humidity

UK curly. CG since October 2013.
Aiming for WL curls (eventually)