A blockchain explorer is a public website that reads a network's records and presents them in plain form: transfers, addresses, balances, fees and confirmations. Each network has its own, so a TRC20 transfer is looked up on a TRON explorer and a bitcoin transfer on a bitcoin one. Using a blockchain explorer needs no account, only a hash or address pasted into its search box. A screenshot proves nothing when the hash cannot be found there, while transfers inside one exchange may never reach the network and appear only in its own history.
To track a bitcoin transaction, copy its transaction ID, also called the hash, from the wallet's history and paste it into a bitcoin explorer. The result shows whether the transfer is still waiting or already confirmed, how many confirmations it has gathered and what fee was paid. Confirmation time depends on how busy the network is and on the fee offered, so a low fee can leave a transfer waiting for quite some time. Searching an address instead shows its balance and every transfer, open to anyone who has it.