What could cause a thumping noise from the front end when turning at low speed?

SOURCE: Hyundai 1998 Elantra Growling sound from front

You have a bad front wheel bearing on the left side because you say when you turn right the noise goes away, because when you turn right the suspension lifts up on the opposite side and releives some weight on the bearing causing it to quiet down. No it is NOT a DIY repair unless you have some pretty extensive tools. the quickest easiest and least expensive way to fix this is to go to a wrecking yard and buy a steering knuckle for the left. it will have a wheel bearing already pressed into it and all you have to do is replace the whole thing and you are done. do not open any brake lines or hoses to do this job just gently press the brake caliper piston back about 1/2" so it will fit over the brake pads when you reinstall.
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SOURCE: Thump sound when applying brakes. No, shaking with

check to make sure you dont have something loose in the trunk or under your seats thats rolling around also check your brakes your rotors may be bad or you have a weak caliper.

SOURCE: Thump when applying brakes. No, vibration from steering wheel.

have you had your under car checked lately ???
take it to local garae get an oil change since they are cheep
once car up in the air tell him the problem they will find it probably and you can call arround for estaments or the parts tell him to look at the bushings they usualy thump if it is a twang then start looking at the springs but they will usualy only twand in turns lock to lock '
also if getting when going over bumps have them look at strut bushings