A bad o2 sensor would not cause white smoke from your exhaust.
A bad o2 sensor could however cause black smoke from the exhaust because a bad o2 sensor could cause the engine to run rich, stall, run rough etc.
Bad fuel can sometimes cause white smoke from the exhaust and also clogged fuel injectors and if so the engine will also likely run rough, could stall etc.
White smoke is also an indicator of burning engine coolant which is not good so make sure you check your coolant level.
If your car is emitting white smoke from the exhaust then you should not continue driving it because it's usually because your car is burning engine coolant.
Your engine will overheat and your engine oil will get coolant in it.
You can pour a bottle of head gasket sealer in your radiator and run the engine for awhile and it should help seal the coolant leak if it's only a slight head gasket leak.
White smoke from exhaust is an indicator of your car burning coolant which means the coolant is getting into your engine cylinders and likely through your head gasket.
You can pour some head gasket sealer into your radiator and it will fix small coolant leaks from head gaskets.
But for larger head gasket damage the entire head gasket or head gaskets need replaced.
White smoke coming from your vehicles exhaust is an indicator that you're burning coolant.
When your coolant is being burnt by your vehicle white smoke will come out the exhaust and it's not a good thing because the coolant is gonna get into your oil.
When the coolant gets into your oil it becomes milky and then your bearings will eventually be destroyed and you'll have a seized engine.
Check your oil and look for signs of milky oil and if so you have a bad head gasket, bad intake manifold gasket, cracked engine head etc.
You said you're losing coolant and that is a sure sign that your head gasket is going bad or has gone bad if you're getting white smoke.
You can go to auto parts stores such as Oreillyauto and buy some Bar S Block seal that you pour directly into the radiator with the coolant.
Then run the engine for 15 minutes at idle with your heater on high.
Then shut the engine off and allow it to cool and check your coolant level and top it off.
I've had good luck doing this and keep a few spare bottles of the block seal in my vehicle with me.
If I notice white smoke coming from the exhaust I immediately dump the bottle of the block seal into the radiator and it stops the coolant burning within a few seconds of the engine running.