Cam Newton scored the most rushing touchdowns by a quarterback in his first 4 seasons, with 33 touchdowns.

which qb has the most rushing td in their first 4 seasons
NewtonAllenCulpepperStewartPrescottGroganLujackJacksonGrahamWatsonBradshawKempMurrayGarciaMcNabbMcNairCunninghamVickZornDouglassKilmerLuck33252222212121191917161615141414141313131312
NAMEGPRUSH TDRUSH YDSATTRUSH AVGRUSH YDS/GREC YDSRECTRGREC AVGREC YDS/GREC TD
Cam NewtonCam Newton62332,5714675.541.5331133.00.50
Josh AllenJosh Allen44251,5623005.235.5121212.00.31
Daunte CulpepperDaunte Culpepper44221,5012695.634.10110.00.00
Kordell StewartKordell Stewart64221,5763035.224.6130101613.02.01
Dak PrescottDak Prescott64211,2212415.119.10010.00
Steve GroganSteve Grogan57211,3702325.924.0000.00
Johnny LujackJohnny Lujack45217421335.616.516116.00.40
Lamar JacksonLamar Jackson46192,9064826.063.20020.00
Otto GrahamOtto Graham48194471752.69.3000.00
Deshaun WatsonDeshaun Watson54171,6773075.531.16116.00.11
Terry BradshawTerry Bradshaw51169711775.519.0000.00
Jack KempJack Kemp38164261203.611.2000.00
Kyler MurrayKyler Murray32151,3632266.042.60000.00
Jeff GarciaJeff Garcia61141,2522624.820.50000.00
Donovan McNabbDonovan McNabb54141,8842786.834.9-136-0.30.00
Steve McNairSteve McNair45141,4402206.532.0000.00
Randall CunninghamRandall Cunningham49141,8742647.138.2-31-3.0-0.10
Michael VickMichael Vick43132,2233047.351.716000.40
Jim ZornJim Zorn56139561825.317.1000.00
Bobby DouglassBobby Douglass40131,6822387.142.1000.00
Billy KilmerBilly Kilmer39137711405.519.8000.00
Andrew LuckAndrew Luck55121,1012225.020.00000.00
Russell WilsonRussell Wilson64122,4304115.938.0171117.00.30
Tim TebowTim Tebow35129891975.028.30010.00
Mark SanchezMark Sanchez62123421252.75.50000.00
StatMuse has season-level data for rushing touchdowns going back to the 1921 season.

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