Having a stop-over makes it multi city. You are traveling to more than one city as opposed to one way or round trip when you are traveling via another city. Price will probably be sky-high unless the fare you are looking at allows stop-overs. Otherwise its just pricing as 3+ one way flights (obviously less taxes, as many are colleted on a per trip basis.)
Enter the following on each line. BKK-NRT, NRT-JFK/EWR, JFK/EWR-RSW/SRQ