$5 PLHE 6-max: ATo BTN vs BB turn decision

pokerstars - $0.05 NL (6 max) - Holdem - 4 players
Hand converted by Holdem Manager 3

BB: $5.18 (103.6 bb)
CO: $8.67 (173.4 bb)
Hero (BTN): $5.00 (100 bb)
SB: $6.22 (124.4 bb)

SB posts $0.02, BB posts $0.05

Pre Flop: (pot: $0.07) Hero has
fold, Hero raises to $0.15, fold, BB calls $0.10

Flop: ($0.32, 2 players)
BB checks, Hero bets $0.15, BB calls $0.15

Turn: ($0.62, 2 players)
BB checks, Hero bets $0.35, BB raises to $1.12, Hero calls $0.77

River: ($2.86, 2 players)
BB bets $2.88, Hero folds

Results: $2.86 pot ($0.12 rake)
Final Board:

BB wins $2.74

Hey no much info on villain 31/19 after 16 hands
1. What flop and turn sizings are the most optimal?
2. I need 27% equity for a call on a turn but do I have that much?
Thanks

Fold turn 100% of the time. Don't worry about being exploited in that spot at 5NL. That's a std. turn fold if you're CR after betting two streets and only have TP ok kicker. At higher stakes you're mostly checking that turn.

Agree with John would even consider checking turn on the level played. What are you trying to accomplish with your bet is it a bluff is it a value bet?