Since train options look bad according to a previous answer, is there a direct bus from Bucharest, Romania to Istanbul, Turkey--preferably an express bus?
Well, if train is not an option you can take a bus. This company: http://marina.autogari.ro/index.php runs buses from Bucharest at 17:00, arriving 05:00 next day. Be warned however that if you find trains a bad option then you might not like this bus either. Here is a testimony from another site: http://www.turkeytravelplanner.com/go/Istanbul/Transport/bus_bukres_ist_bad.html I take it with 'cum grano salis' myself, but then again i don't see a problem with a train either. I can't find any info about Bucharest, but i know from first hand that Turkish METRO company runs daily buses from Ruse on the border with Bulgaria that cost about 60 Bulgarian Lev(30 Euros) and I had nothing but great experiences with them.
Maybe the best option for you would be a Pegasus Fly company. They are a Turkish budget airline that flies daily from Bucharest to Istanbul and tickets can often be found for 40-50 Euros if booked in advance. http://www.flypgs.com/en/
Without local knowledge, it's hard to answer, but, the internets can come to your rescue. It seems you have to switch buses in Ruse:
From local internet search I see there are bus options from İstanbul, I think it takes about 12 hours. You may want to check this address they have Romania telephone numbers for reservation as well.