AQs squeeze in BB

This is a zoom hand, villain is unknown.

I was deciding whether to barrel turn or not, decided to check back and take the free card. Should I be barreling turn because this kind of flop will have a lot of people calling, and also because we still have two overs and the nut flush?

Is this river also a standard check back after not barreling turn and failing to improve.

Hand converted with online PokerStrategy.com hand converter:

Play hand

$0.01/$0.02 No-Limit Hold'em (6 handed)

Known players:
[tg]BU$2.00:///////;SB$2.91:///////;BB (Hero)$2.09:///////;MP2$2.08:///////;MP3$3.75:///////;CO$2.00:///////;[/tg]

Preflop: Hero is BB with A:s, Q:s.
2 folds, CO raises to $0.06, BU folds, SB calls $0.05, Hero raises to $0.26, CO folds, SB calls $0.20.

Flop: ($0.58) 4:d, 3:s, 9:s (2 players)
SB checks, Hero bets $0.30, SB calls $0.30.

Turn: ($1.18) T:c (2 players)
SB checks, Hero checks.

River: ($1.18) 3:c (2 players)
SB checks, Hero checks.

Final Pot: $1.18.
Results follow:

Hero shows a pair of threes(A:s Q:s).
SB shows two pairs, tens and threes(8:s T:s).

SB wins with two pairs, tens and threes(8:s T:s).