Home winger Catalin Fercu scored two first-half tries but his one-man show couldn't prevent Turner's men from recording a crucial bonus-point victory.
Turner said: "We're very disappointed. Every time we scored we leaked a try. We've won with five points, which is what we set out to do, but we never do anything easy.
"We've got a few bruised egos today and I think the players will have more respect for Romanian rugby after this. Bucuresti gave us some game."
HT: 10-24 Attendance: 500
Bucuresti: D Dumbrava (A Barbuliceanu 52), C Fercu, C Dascalu, C Gal Minya, S Ciuntu, I Dimofte (D Vlad 62), A Lupu (V Calafeteanu 57), N Nere (F Dobre 60), M Mihalache (capt, M Adascalitei 70), P Ion, N Rus, C Tatu (M Baraulea 52), S Burcea, M Macovei, C Mersoiu (D Carpo 67)
Replacements used: F Dobre, M Adascalitei, D Carpo, M Baraulea, V Calafeteanu, D Vlad, A Barbuliceanu
Scorers: Tries - C Fercu (2), S Ciuntu, D Vlad, A Barbuliceanu; Cons - D Dumbrava, A Barbuliceanu
Yellow cards: P Ion (30), N Rus (59)
Newport Gwent Dragons: K Morgan (capt), G Wyatt (A Brew 67), P Emerick, A Smith (M Hercus 78), N Brew, C Sweeney, G Cooper (G Baber 62), A Black, K Crawford (R Wilkes 77), L Harrison (J Corsi 77), P Sidoli, L Charteris (I Gough 45), M Owen, C Charvis, N Fitsemanu (J Ringer 45)
Replacements used: R Wilkes, J Corsi, I Gough, J Ringer, G Baber, M Hercus, A Brew
Scorers: Tries - A Black, C Charvis, P Emerick, G Baber, A Brew; Cons - C Sweeney (4); Pens - C Sweeney (2)
Yellow cards: J Ringer (53)
Referee: Carlo Damasco (Italy)