Can I begin a turn when there is only 1 card remaining in the player deck?
The game ends when you have to draw cards and there are not enough.

If, as you are about to draw, there are fewer than 2 cards left in the
Player Deck, the game ends and your team has lost! (Do not reshuffle
the discards to form a new deck.)

But you perform your actions first:

Each player turn is divided into 3 parts:

Do 4 actions.

Draw 2 Player cards.

Infect cities.



Reference to the rules.
So even if the deck is empty, if you win the game before the draw phase, you win.