Just adjusting the nonce is not sufficient for mining a valid Bitcoin block at the current difficulty level which is around 80 exa-hashes (80 x 1018 hashes) per second. Nonce field in the block is only 4 bytes which means it can at most provide you 2^32 different hashes(4.2x109 hashes). That is the reason, miners adjust the extra nonce value in the coinbase transaction and then iterate over the entire 232 nonce field again.
Supercomputers are way slower than ASICs for mining purposes. Supercomputers contain huge CPU power that are highly useful for general purpose computing however, for running a particular task they will be much slower compared to a hardware that is designed specifically for that purpose. That is the reason miners use ASICs where the
dsha256 algorithm is printed on the circuit in the hardware. These chips can't do anything else, but run the double SHA256 hashes which provides them advantage over even a supercomputer.
Miners generally run another machine connected to these ASICs chips. While the ASIC chip is exhausting the nonce space, the other machine changes the extra nonce value and queues it up for the ASIC hardware to take it from there. So these task can be run in parallel.