Good answers here. As all of them suggest to download ordered items and manage your orders in an offline manner.
Another way is to, keep track of your collection (sets and loose/extra bricks) online in Rebrickable. Here you you have an option to import orders directly from Bricklink.
Click the Import/Delete Parts button on the Part List page that you want ...
To add to @user18321’s answer, here are two other ways:
Within BrickLink: go to each order and click on “Add to Wanted List”. Now add the pieces to either a new wanted list (perhaps under the order number?) or dump all your orders into one big Wanted list (e.g. “Purchases”).
Using BrickStore: download this free software, give it your BL credentials, and it ...
If you go to your order overview, you can click on Download
On the next page select the checkbox 'Include detail items'
This should give you and overview of all orders including exact parts bought in each transaction.