Galton opened the scoring on 26 minutes with a long-range shot before adding to her tally on 63 minutes with an angled shot.
Turner scored United's second in the dying moments of the first-half when she headed in a corner.
Sigsworth's goal also came from a corner, an area that United manager Casey Stoney has said she wants to see her team score more from, where she picked up a loose ball on 52 minutes.
Salmon got a consolation goal for Bristol on 72 minutes when her first shot rebounded back to her and she was given another opportunity to lash it home.
The last laugh went to Heath and United, however, with the USWNT star scoring twice in the final 10 minutes.
Her first goal was a left-footed shot from outside the box on 83 minutes and her second came three minutes later when she pulled the ball behind the Bristol defence and slotted it into the net.
The win means United sit top of the WSL table unbeaten in 14 games, four points clear of nearest rivals Arsenal.