No, transaction fee can not consume half of the bitcoin sent, it depends, bitcoin transaction is measured in weight. For example, you want to make a transaction of $10 with segwit address, while the mempool is not congested and a feerate of 30 sat/vbytes can make the transaction included in the next block and be confirmed, this will cost around $1.8 with the present bitcoin price, also consider the transaction to have 1 input and 2 outputs. Although, $1.8 is paid for $10 transaction, but if it is the same constant (1 input, 1 output, segwit address used and mempool is 30 sat/vbyte), making higher transaction like $1000 or even higher will still only cost $1.8 in fee to be paid, but this is when you use wallets.

Transaction fee is not determined by the amount of bitcoin paid, it is determined by:

• Mempool
• The type of address used
• Number of inputs and Outputs
• Transaction weight

If mempool is not congested, you will pay lower fee, the more the congestion, the higher the feerate and the higher the fee. The number of inputs in a transaction makes the transaction weight heavier, which will significantly increase the fee, the number of output also does but does not make a transaction weight much heavier if compared to number of inputs.

On exchanges, the fee is not flexible but static, exchanges have a constant fee customers do pay. Like on binance, the fee is 0.0005BTC, likewise on HBTC exchange. On HitBTC, the fee is 0.0009BTC, there are some exchanges that still charges higher and some charges lower fee, but on each exchange, the fee is constant irrespective of the addresses used. Although, the exchanges uses this to gain because the exchange will pay less fee to miners, use less fee to consolidate and gain the remaining fee paid. So, the extra fee which makes exchanges fee higher and yet constant is for the exchange maintainance, and this is not how normal wallets operates.