Does the draw from Sylvan Library count as your normal card draw for the turn?

Sylvan Library's triggered ability says you 'may draw up to two additional cards' at the beginning of your draw step. The word additional is the key: it means on top of the one card you already draw normally during your draw step (CR 703.4, 121.1). So in total, Sylvan Library can let you look at up to three cards per turn cycle — one from your normal draw plus up to two extra from the trigger.

After drawing those extra cards, you must put two cards from your hand back on top of your library for each additional card you drew (CR 603.6 governs triggered abilities resolving; Sylvan Library's oracle text spells out the payment: pay 4 life per card you keep). So if you draw zero extra cards, you pay nothing and keep just your normal draw — the trigger did nothing meaningful.

Importantly, the draw from Sylvan Library's trigger does NOT replace your normal draw step draw. The normal draw happens first (CR 504.1), and then, at the beginning of the draw step, Sylvan Library's trigger goes on the stack and resolves, offering additional draws (CR 509.1). Both events are distinct.

Concrete example: It's your draw step. You draw your normal card (say, a Forest). Sylvan Library's trigger then resolves; you choose to draw two more cards (a Counterspell and a Llanowar Elves). Now you must either put two cards back on top or pay 4 life for each you keep. You put Counterspell back and pay 4 life to keep Llanowar Elves. Your hand nets one extra card at the cost of 4 life.

In summary, the normal draw step draw (CR 121.1, CR 504.1) is entirely separate from and unaffected by Sylvan Library's triggered ability. Sylvan Library supplements but does not interfere with your mandatory draw.