A stabilizing apparatus for paper webs, comprising at least a movable element (4) disposed in correspondence of a paper web (2); tho movable element (4) comprises a wedge-like portion (5) having a lower surface (7) for intercepting an entrained first fluid thread (F1) in correspondence of a lower face (2a) of the web (2) and diverting said first thread (F1) along a direction transverse to the advancement direction (A).