Because women, wine, and books cost money, and Tyrion doesn’t have any outside of his family. It goes back to what Tyrion said when he and Jon were on the Kingsroad to Castle Black: “if I had been born a peasant, they might have left me to die.”
Tyrion doesn’t fit very well into the economy of a feudal society – without an inheritance from House Lannister, he’s not going to have an estate that provides him an income. He’s not physically capable of performing the labor that makes up 90%+ of the labor market (although a market in labor doesn’t really exist at this time). He’s literate, but he’s not a maester, so he’s got a credentials problem. He’s a skilled politician and administrator, but given that those positions are filled based on blood ties and/or the patronage of liege lord to vassal, as a political nobody he’s not going to get an appointment as a steward or anything.
At best, Tyrion might be able to eke out a living as a scribe or clerk in some merchant’s office, but he’s not going to be able to afford much of wine, women, and books on a clerk’s salary.
In ADWD, we see how precarious the life of a dwarf on his own can be with how quickly and easily Tyrion is thrown into slavery, and from the example of Penny’s precarious existence. It’s only by extreme luck and his own cunning that he’s found a place in a mercenary band rather than dying in a fighting pit as an undercard amusement.
Braavos seems pretty meritocratic, especially for the setting.
Given their roots as a melting pot of escaped slaves, wouldn’t the Braavosi welcome immigrants, especially ones as skilled as a noble Westerosi? I’d think the Iron Bank would have particular use for a Lannister with knowledge of both Casterly Rock and King’s Landing, and the major players therein.
I’m sure Braavos does welcome immigrants, but it’s also an intensely capitalist society, so they especially welcome immigrants with money. Look how quickly Sam, Aemon, and Gilly went from being OK to poverty.
So while Tyrion could do well there if he had starting capital to pay for room and board, transportation, bribes and other investments, it would be very difficult indeed if he showed up with just the clothes he stood up in.