Originally Posted by irishguy28

Turkish Airlines will not allow you to board the flight to Istanbul if you do not have the necessary visa to enter the country (unless you are somehow entitled to obtain such a visa upon arrival). The fact that you intend to immediately depart on a separate ticket makes no difference.

Because you booked a ticket that ends in Turkey, TK are compelled to ensure that you comply with the Turkish entry requirements applicable to holders of your passport. If you don't meet the entry requirements for Turkey, you risk being left in Budapest.

Additionally, given that you are on separate tickets, the airlines will almost certainly refuse to interline your bags. You MUST plan on entering the country, retrieving your luggage, and then checking yourself and your luggage in for your next flight.