I've just bought a new (over 40yrs old) home. It doesn't have A/C and
it becomes very warm during the day (+85F)

I've installed screened windows & doors (to allow for circulation) but
it still becomes very warm. I think the attic (crawl space) is
trapping heat. I don't have any exterior ceiling ventilation except
for soffit vents.

If not this, what else could be causing the heat to build up? How
best to fix?