What is the best/cheapest way to get fomr Bodrum to Istanbul, Turkey?
Because of the geography and condition of the roads (yikes!), air is your best choice.
Atlas Jet flies Bodrum to Istanbul Ataturk. I flew Atlas Jet last October from Izmir to Istanbul. Nice flight on a newer Airbus jet. Booked online a few months in advance and got a great price - well under $100 for a one-way ticket.
you want to fly. There may be big bus goes from bodrum to istanbul, but the one i took from bodrum to selcuk was about 3-4 hrs (bad road), and from selcuk to istanbul is overnight about 8 hr, so with bus it's at least 12 hrs if not more. if you have time to visit selcuk (ephesus) then bus will work, or you can train from selcuk to izmir then fly.