How to color a curve
Your plot
c = ContourPlot[y^2 - x (x - 1) (x - 3) == 0, {x, -2, 7}, {y, -4, 3}]
The region you would like colored
R = RegionPlot[y^2 - x (x - 1) (x - 3) < 0 && x < 2, {x, -2, 7}, {y, -4, 3}]
Both plotted together
Show[c, R]
Try without "==0"
ContourPlot[y^2 - x (x - 1) (x - 3), {x, -2, 7}, {y, -4, 3}]