Cheap tyres, or a mix of tyres with different wear, can have a very detrimental affect on roadhandling Moggie. Continental 185 tyres were notorious for losing grip, even when the road wasn't wet, yet their 195s transformed the same car when fitted. Continental acknowledged the 185s were a mistake.
Tread depth is just one factor of tyre safety. Just as important is the age of a tyre as the rubber deteriorates over time and old tyres can affect handling quite noticeably. Indeed, the MoT is being changed soon to fail any tyre over three years old - regardless of tread wear.
Tyres are the most important safety devices on the car and you should never cut corners on price. Always avoid budget brands and always fit four new tyres together if possible as they will then wear at the same rate and have the same roadhandling properties. At least always fit tyres as a pair on the same axle - never just one tyre. If you can change your tyres like that you will really notice the difference in handling. That's probably why you car handles badly now.
Although some people will try to save a few pounds on tyres they will then happily spend three or four times the prices of new tyres of sound systems and other bolt-on goodies! Does that make sense?