There's no clear line between "thermonuclear weapon" and "basic fission weapon." Indeed so-called "thermonuclear" weapons get most of their yield from fission reactions - they are more appropriately called "fusion-boosted fission" devices. You could also call them "hydrogen bombs," if you liked, as they use a hydrogen isotope to boost neutron flux and get a bigger fission yield. But you don't need two stages to boost fission yield with hydrogen fusion - this was done with single stage bombs first, by injecting tritium into the center of the pit. You can scale it much larger with two stages, but it's not a fundamental difference in the principal of the mechanism.

If "thermonuclear" bombs worked in the manner one would think they work from the term, then they would indeed be a whole different beast - if they used a fission weapon to "ignite" fusion fuel with incredible heat. This is not how they work. They work in precisely the same way as pure fission weapons, through the compressive action on the nuclear fuel in which heating the fuel is actually undesirable and reduces efficiency. The difference is that "thermonuclear," or correctly labeled "two stage fusion boosted fission" bombs, use the ablative action of the radiation from the bomb casing to compress the second stage. The second stage's purpose is to create very high neutron flux and thus induce fission to produce most of the bomb's overall yield. If you can call that "thermonuclear" you can also call Little Boy and Fat Man "thermonuclear" - they also use heat-driven ablation to compress nuclear fuel enough to create a reaction.

To further blur the lines, we use small linear accelerators to induce fusion and thus neutron flux and the start of the fission chain reaction at the precise ideal moment to ignite our compressed fuel in the first stage (which then compresses and ignites the tritium stored in its center, boosting the fission reaction through fusion-produced neutrons.)

So the whole stage is electric ignition of chemical explosive, compression of fission fuel to ideal form factor, fusion through electric linear acceleration to produce the first neutrons to ignite the fission fuel, compression of the hydrogen isotope tritium in the center of the first stage to produce more neutrons to boost the fission reaction, heating of the bomb casing causing re-radiation of x-rays to the second stage, 'z-pinch' of the second stage through x-ray ablation compressing the lithium-hydrogen mixture in the cylinder of the second stage, ignition of said mixture producing huge flux of fast neutrons, and (this is where most of the yield comes) ignition of the second stage casing and center fissioning 'spark plug' by fast-neutron flux.

That's the state of the art at the moment. As you can see, it utilizes both fusion and fission at different intervals throughout the multi-stage process. Even only in a single stage, fusion is utilized twice to boost fission yield/efficiency. The second stage is more of the same, using fusion to boost fission on a larger scale. But if you perfect the first stage, you already have quite an efficient, formidable weapon, and have demonstrated mastery of all the concepts and skills you need to go the step further. You have already gone from using pure fission fuel, to using fission and fusion and fusion-boosted fission, to using fusion ignition and fission and fusion and fusion-boosted fission. You need to go from that to using fusion ignition and fission and fusion and fusion-boosted-fission and fusion and fusion-boosted fission. That's a step, but it's not a huge step for someone who is already almost there, and knows both that it is possible and the principles of modern design.

Where is North Korea? Probably at Fat Man stage, moving forward. The low yield from their initial tries was probably from too-high levels of plutonium 240, which likes to fission spontaneously and ignite the fuel before it is in the ideal configuration. They need further refinement and/or better luck to get the yields the Manhatten project got on its first tries. The jump from where they are to state-of-the-art is indeed quite high, but that's not a jump they have to make. Instead, they have to make a series of much smaller jumps, just like we did - and they are probably on their way up.