Well, this could be scary. You hit 60 or 70 ish miles an hour and all of a sudden you're getting a lot of feedback in the steering wheel. The car no longer feels safe on the road. There are really not that many possibilities, so let's run through them together.
TIRES. Your tires are what connect you to the road. If they are out-of-balance or maybe have never been rotated, that is the first and easiest culprit to check for. You can take your car to the nearest tire shop and they should be able to check if they are properly balanced and look for any irregularities in wear (funny wear patterns).
SUSPENSION. Your suspension components wear over time. Many of these components contain rubber and grease and therefore need to be replaced when worn. If your suspension is worn it will also be causing tire wear.
ALIGNMENT. All of those above suspension components need to live together in harmony (in alignment) in order to drive straight down the road. Photo from https://www.wapcar.my/news/wheel-alignment--what-is-camber-caster-and-toe-2645
STEERING. Usually for cars, there is a part called the rack and pinion. For bigger vehicles, there are a couple different types of steering set-ups. All steering setups have components that wear (usually joints or bushings, or seals). This can cause feedback in the steering wheel if any of these components are loose or worn.
PREVIOUS DAMAGE. Lastly, if you had previously had repairs performed (if the car had been in an accident) it could be possible that you have damage that hasn't been repaired.
Make sure you feel safe on the road and start with the easiest and most important thing first! Your TIRES!