Simple rules to reduce thickness:
Choose a frame with centres approximately equal to (or slightly less) than your pupil distance.Frames have a measurement such as 52 - 18 - 135. The 135 is the arm length which you can ignore. The important numbers are 52 - lens diameter and 18 - bridge width. In this case if indicates that the lens centres are idea for a pupil distance of 70mm (52+18). If you PD is much less than this, outer edge thickness will increase with a minus prescription. If it is 70 or more in this example it will reduce the apparant thickness of the lens.
Choose a frame with a smaller lens diameter.
Choose a higher refractive index lens. Plastic lenses go up to about 1.74 but are costly. Glass lenses can get up to around 1.9 but are eye watering in price and can be fragile and heavy. Ignore any brands for single vision lenses. Quality of finishing from Essilor, Hoya, Zeiss, Kodak are all excellent. There are no significant differences from a customers perspective.
Unless they really don't suit you, get a plastic frame. It will hide the edge.
Ask the opticians to edge polish the lens. While it can increase internal reflections, this will be offset by the antireflective coating and will reduce the apparant of the lens thickness.
I hope that helps. I might be an accountant now but I spent a few years working in the industry and had my own business supplying glasses to independent opticians. Let me know if you want any more advice.