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I believe (and I'm by no means an expert at this game) that your problem is that your naval stockpile slider is too low, meaning that the ships aren't getting all the supplies they need. This, in turn, means that they take attrition when in port. However, if I'm wrong on this point, someone feel free to correct me.
 
I'm guessing that you have no clipper convoys stockpile and haven't unlocked reform letting you build navy base? I think BenandorfGaming as Japan or Shenryyr2 as China had similar problems in their let's plays.

Clipper Convoys at the start are in high demand and low supply, you as unciv with low prestige get their needs met last before other nations.