There was Dorne ships if you look again. Which I guess was sailed over to help transport troops.
And Tyrell ships, which could be seen as well in the scene.
Be extremely generous and bump it to 600 ships altogether even. That would still require almost 200 men and horses per ship. If they were going with this route, the ships just have been portrayed a lot bigger.
Considering the way the show did it compared to the books. One could easily say that the fleet would consist of a 1000 ships and that's without being generous. I mean, the Ironborn fleet, the Tyrell fleet, the Dornish fleet and the fleet of the free cities that she captured combined.
Especially considering it is apparently easy to build a 1000 ships when we look at Euron and how he puts it. Sure like he suggest it would be the biggest fleet (if my memory serves me right), but we're talking about several major houses, plus the navies of a bay that pretty much relies on ships as well. The Reach for example has various houses that have their own considerable navies. And I wouldn't be surprised if the same goes for Dorne.