You're not telling us how much you weight, but pushups require you to push a quite large portion of your body weight, my own unscientific measurements with my hands on a scale in pushup position tell me ~60%.
Doing 13 kg in bicep/tricep workout doesn't really tell us anything. Different types of exercises vary a lot in how heavy they are due to the different mechanics of the exercises. I can deadlift about 160 kg, but I'm struggling with 8 kg in standing shoulder flies.
Regarding compound lifts, the fact that they use more muscles doesn't mean that the muscles share the burden, it means that the movement is more complex and requires a larger amount of work in total. It's like a chain, every link has to be strong enough, that's why they are popular.
My recommendations would be to do pushups on your knees, negative pushups (just do the lowering, first half of the movement, slowly, you are 40% stronger in the exentric phase than in the concentric so it should work better), dumbbell presses and planks.