Bad news: cycling is an expensive sport.
It might be the case that buying maximum one size bigger bike, counting on the bigger flexibility of the children and playing with stems (they are not so expensive), a bike may be used for 3 years. So: 13+3+3=19. The third bike would be the last that is constrained by the age.

I guess that renting is not a big deal, because it takes some more time to get used with a new geometry. Besides... cycling requires continuous training, like any other sport, so how can somebody get trained on a long run without a proper bike?

I assume that a 300 $ bike should be ok for 13 years old, to get used with the sport. At 16 years, a serious junior would probably need something at least 1000 $.

I made a lot of research over the internet for buying bikes and my assessment was that the price of a very good bike is around 2000 - 2500 $ (but even 1500 - 2000 $ is more than acceptable) and everything above that price brings very little to nothing to the performance.
Apart from the price, the type of bike also plays a role: a time trial bike is somehow different versus a road one (aero position). But this is for really performance oriented people.