in R, plot a nonlinear curve

There are several references that come close, but my lines() is producing multiple arcs instead of just one nonlinear curve. It looks like a hammock with a bunch of unwanted lines. How do you generate a simple nonlinear line? Dataset available as Auto.csv at http://www-bcf.usc.edu/~gareth/ISL/data.html.

    library(ISLR)
    data(Auto)
    lm.fit1=lm(mpg~horsepower,data=Auto) #linear
    lm.fit2=lm(mpg~horsepower+I(horsepower^2),data=Auto) #add polynomial
    plot(Auto$horsepower,Auto$mpg,col=8,pch=1)
    abline(lm.fit1,col=2)       #linear fit
    lines(Auto$horsepower,predict(lm.fit2),col=4)  #attempt at nonlinear