Well, keep in mind the specific circumstances in which that happened:

“The garrison had been prepared to sell them to the Usurper, but one night Ser Willem Darry and four loyal men had broken into the nursery and stolen them both, along with her wet nurse, and set sail under cover of darkness for the safety of the Braavosian coast.

She remembered Ser Willem dimly, a great grey bear of a man, half-blind, roaring and bellowing orders from his sickbed. The servants had lived in terror of him, but he had always been kind to Dany. He called her “Little Princess” and sometimes “My Lady,” and his hands were soft as old leather. He never left his bed, though, and the smell of sickness clung to him day and night, a hot, moist, sickly sweet odor. That was when they lived in Braavos, in the big house with the red door. Dany had her own room there, with a lemon tree outside her window. After Ser Willem had died, the servants had stolen what little money they had left, and soon after they had been put out of the big house. Dany had cried when the red door closed behind them forever.”

…At first the magisters and archons and merchant princes were pleased to welcome the last Targaryens to their homes and tables, but as the years passed and the Usurper continued to sit upon the Iron Throne, doors closed and their lives grew meaner. Years past they had been forced to sell their last few treasures, and now even the coin they had gotten from Mother’s crown had gone. In the alleys and wine sinks of Pentos, they called her brother “the beggar king.” Dany did not want to know what they called her.

Darry didn’t have time to organize an entourage of Targaryen loyalists, move the treasury, and instead booked it. And while Darry was alive, Dany and Viserys lived the life of a royal in exile, living in a manse with many servants, intriguing with the Sealord and the Martells, etc. 

The downfall started when Darry died and left them without adult supervision and the servants stole their money, leaving them penniless and homeless – note those things are very much linked; with a more substantial entourage of loyal retainers, the looting would have been forestalled and they probably would have stayed in Braavos.

Even when they left, they were taken in by the elite of Essos. However, as with many exiles, once it becomes clear that the new regime isn’t going anywhere, they lost their political cachet and began

to fall into genteel poverty, cushioned by the fact that they still had treasure to spend.