If your turntable uses sintered bronze bushes it is important to use the correct type of lubricant.
Sintered bronze bushes are porous and can be made to absorb oil. This gives them a certain degree of self lubricating ability.
Older automotive gear oils are most definitely a very bad choice as they contain sulphur based EP additives (extreme pressure) which are corrosive to yellow metals. Most modern gear oils contain buffered or deactivated sulphur which is less far less corrosive to yellow metals. However these are still damaging to yellow metals when you have a hard metal like steel in contact with a soft metal like bronze. This is exactly what we have in a turntable such as a Thorens TD124 or TD125.
You might get away with a modern gear oil I wouldn't risk it.
Engines oils are a slightly better option but they do still contain additives which are totally irrelevant in a turntable bearing and may even cause some problems.
I would not use any oils containing PTFE (teflon) or molybdenum disulphide.
The best easily available type of oil to use is an oil designed for air compressors. These oils do not contain additives which will be damaging to bronze bearing, they come in suitable viscosities and they are relatively easily obtainable.
The main advantage of synthetic base stock oils is their ability to withstand higher temperatures without breaking down and there better cold flow properties at the other end of the temperature range. This is of no benefit in your turntable bearing. Having said that the cost difference is pretty negligible in the quantities we are talking about so if you want to buy synthetic then go ahead. It certainly won't do any harm and it may provide some slight advantages.
Compressor oils tend to come is ISO viscosity ratings. Here is a rough guide.
ISO 32 = SAE 10
ISO 46 = SAE 20
ISO 68 = SAE 20w
ISO 100 = SAE 30
If your turntable has tighter bearing clearances a lower viscosity may be better and conversely if your turntable bearing is old and has widened clearances and higher viscosity may be better.