The live script uses the same axes for each plot unless you specifically tell it to create a new figure. So likely what is happening is that, rather than 3 separate figures, each plot is replacing the previous plot.
Try adding a figure command before creating each successive plot
% First plot
x1 = linspace(0,6);
y1 = sin(x1);
plot(x1,y1)
ax1 = gca;
% Second plot
figure
x2 = linspace(0,10);
y2 = 2*sin(2*x2);
plot(x2,y2)
ax2 = gca;
% Third plot
figure
x3 = linspace(0,12,200);
y3 = 4*sin(6*x3);
plot(x3,y3)
ax3 = gca;
linkaxes([ax1 ax2 ax3],'xy')
