I use both the AnkiMobile on my iPhone and Anki on public laptop. Therefore, I will sync my learning progress to ankiweb and download my decks to the public laptop everytime.
My AnkiMobile has no problem in syncing, but it takes so much time when I download my deck to the public computer. I will regularly delete decks so there is only limited number of cards in my AnkiMobile. While when I sync to the public laptop, it downloads hundreds of thousand media, so I wonder are they keeping my psat records.
It sounds like you have old media still hanging around, and you’re downloading that every time. On your iPhone, try running Preferences > Check Media (Preferences - AnkiMobile Manual ) and delete any unused media.
Syncing is intended to sync your database and files to a new device and then sync differences (leaning progress, new/changed notes and files) between devices. Downloading everything again and again for temporary use to this „public notebook“ likely puts strain on the platform.
To @ferophila 's point – Another option for you would be to run Anki as a “portable” app from a flash drive. Your local collection would be stored on the flash drive, but still able to sync normally with AnkiWeb when you open and close Anki. It would also save you a LOT of time. Managing Files - Anki Manual