You can't do this, since you discard immediately if at any time you have more than 7 cards. From the rules:
If you ever have more than 7 cards in hand (after first resolving any Epidemic cards you may have drawn), discard cards or play Event cards until you have 7 cards in hand (...).
Drawing epidemics is the only exception; those don't stay in your hand long enough to cause you to discard.
This is also mentioned under "commonly overlooked rules":
Your hand limit applies immediately after getting a card from another player.
Note that the reminder text in the corner of the board might lead you astray here, saying:
- Take 4 actions
- Draw 2 cards
- Resolve epidemics
- Discard to 7 cards
- Draw Infection cards
You should read that as a reminder that you may need to discard then, not as an indication that you don't need to discard at other times.