Have you tried to use towers (In the support section where you can choose a minimum and maximum width) or conical supports (In experimental where you can choose degrees as well as minimum widths)? Depending on which version of Cura you use, it may or may not be available.
You can see it (Them) in action here:
Some other slicers do this. I think someone said maybe stratasys slicer does this. It should actually be more hourglass shaped where it is strong at the base and then gets thinner with height and then widens only at the last few mm.
Personally I design most of my supports myself in cad.
There are some great programs out there that do better support than cura and then saves it as an STL. Consider this one:
pay particular attention to how to rotate your part in the "annoyances & limitations" section and also note that there is a small error where he set layer height to "0.5" mm and it should be the layer height you print at e.g. .2 or .1mm.
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