How am I supposed to go to conferences with a reimbursement system as a poor TA?
This is an unfortunate reality of life. I think at one point in grad school I had $5,000 in outstanding reimbursements because it took so long to process them. Many grad students don't have the liquidity to handle that.

You can, and should, ask whoever will reimburse you whether (i) they can book for you, (ii) let you use their payment card to book, or (iii) pay you in advance. It's often possible and there's no harm in asking. The situation is common enough that they will be used to such requests. For example, people traveling for academic job interviews will rack up a lot of travel expenses in a short time and will sometimes ask to have costs paid directly.

Unfortunately, some places have rigid bureaucracies that insist on operating through reimbursements or some other arbitrary policy. (I recently had a trip where they insisted that they book the flight for me, that I get the hotel reimbursed, and they gave me a per diem for food. Three different policies from one bureaucracy for one trip!) Ultimately, there is little that can be done about it if that's the case.

In the long term, the best solution is to save up a buffer to cover expenses like this. Alternatively, you could get a credit card or other loan to pay for it. However, that is more of a question for https://money.stackexchange.com/