Yes, I'd say from the contracted, underrun heels and the ratty look of the frog, it is very likely your horse also has thrush, but it's hard to critique a foot still covered in mud! I don't think shoeing = thrush or vice versa. It happens regardless of shoes & is more about the environment & general health of the feet IME.
Thrush *commonly* smells bad, but not always. Don't rule it out just because his feet don't smell septic. You can treat the infection, as others have suggested, with a range of things. ACV often works, but t-tree or eucalyptus oil are more 'broad spectrum' antiseptics.
But the infection is more of the 'symptom' IMO. As mentioned, environment is a big factor & keeping him out of mud & on dry footing as much as possible is important. Working towards getting those feet into healthy, well functioning form, so he can start using & therefore developing his heels is another thing. Diet - low GI diet & well balanced nutrition is another big factor of hoof health.