Typically, booking a whole trip with different airlines would be risky. For example, let's say your first flight with JetBlue was delayed, and you missed your connecting flight with Norwegian Air. Because the airlines have no association with each other, Norwegian Air has no obligation to reschedule your flight for free, so you would just lose your money. Kiwi.com instead offers their own guarantee, which covers schedule changes, flight delays, and cancellations. As long as you contact Kiwi.com as soon as you're aware of the delay, they will provide you with an alternate connecting flight, or a full refund, at your discretion. We haven't used this guarantee ourselves, but it certainly sounds like an appealing way to take the risk out of a thrifty flight hack!
Years ago, if you wanted to fly between continents, you were mostly stuck with traditional expensive airlines. That’s no longer true. Budget airlines now service many long-haul routes, making it possible to fly around the world for very little money. Norwegian Airlines allows you to fly between Europe and Bangkok for about $250 each way. AirAsia offers crazy-cheap deals around Asia and Australia for as little as $100 each way. Indian and Middle Eastern airlines offer cheap flights throughout the subcontinent and Africa. You can fly most of the way around the world on a budget airline!
You're not crazy for thinking that a flight price has changed after searching it a few times in your web browser. Based on the cookies in your browser, flight prices do increase when a particular route is repeatedly searched, as the site wants to scare you into booking the flight quickly before prices get even higher. Always search for flights in incognito or private browsing mode to see the lowest prices.