Bike HEADSET Care...
does your bike have a threaded headset?
If the Steering tends to follow "lines and grooves", and feels stiff or notchy, then the Headset is either way too tight or the grease has failed... the first option is fixed by SLIGHTLY loosening then re-tightening the Top cup and Top Nut of the headset, and ensuring that the steering now works smoothly and that there is no "Clunky" looseness of the headset... the Second option is a complete servicing the headset by disassembling , cleaning, inspecting , CAREFULLY AND THOROUGHLY, for Any Wear or rust pitting, then replacing the assembly if rusted or pitted...

the second option should be done only by a competent mechanic . Both of these "Threaded Headset" operations are shown in videos on YouTube...

if the bike has the more modern "Non-Threaded" Steering stem and Straight steerer tube/fork... the tension applied by the Top Cap/screw sets the preload of the steering bearings in the Headset, and will require that a specific procedure be followed... check for drag/roughness/ clunky operation of the steering... Problem d=found? loosen top cap, loosen steering stem clamp, SLIGHTLY tighten top cap screw until no "clunky" is felt, but no drag is present... Tighten Stem clamp screws to proper torque(OVER-Tightening will cause possible breaking of the stem, and possibly your face, too!) then i loosen/re-torque the top cap screw.

you can also find this procedure on youtube.

if any of these procedures cause you to worry... take the bike to a shop for repair, ok?

i hate seeing anyone getting hurt.

oh, and there MAY be a bad Frame bend or break causing the erratic handling too.
no one has mentioned THAT, from what i've seen.
I'd seriously doubt it's got anything to do with the tires themselves ..
other things to consider... poorly centered wheels... a bent Fork... badly cut Dropouts causing wheel misalignment(found on New WallyMart-quality bikes, all too often)... badly Installed wheel(s).... and, another WallyMart thing, the Backwards Suspension fork (fixed one of those, two weeks ago)