Vitamin C bioavailability
According to the review Synthetic or Food-Derived Vitamin C—Are They Equally Bioavailable? (Nutrients, 2013), the bioavailability of vitamin C from foods and supplements is similar:
...all steady state comparative bioavailability studies in humans have
shown no differences between synthetic and natural vitamin C,
regardless of the subject population, study design or intervention
and, according to Institute of Medicine (in the US):
The type of food consumed has not been shown to have a significant
effect on absorption of either intrinsic or supplemental vitamin C.
Vitamin C supplements as prevention for common cold
Vitamin C supplements, even in doses 200+ mg/day (more than 3 x recommended dietary allowance - RDA) do not likely help in common cold:
This review is restricted to placebo‐controlled trials testing 0.2 g
per day or more of vitamin C.
Twenty‐nine trial comparisons involving 11,306 participants
contributed to the meta‐analysis on the risk ratio (RR) of developing
a cold whilst taking prophylactic vitamin C.
The failure of vitamin C supplementation to reduce the incidence of
colds in the general population indicates that routine prophylaxis is
not justified. Vitamin C could be useful for people exposed to brief
periods of severe physical exercise. (Cochrane, 2007)
To get 200+ mg vitamin C from citruses, you would need to eat at least 3 oranges or 7 lemons.
Considering the above evidence, the improvement of cold symptoms was likely a natural process.
Bioavailability of other nutrients from foods/supplements
There is no general rule to say that nutrients from foods or supplements are absorbed better or worse; it can depend on a specific food and a specific supplement formulation.
In our in vitro model, naturally iron-rich mineral waters and
synthetic liquid iron formulations have equivalent or better
bioavailability compared with ferrous iron sulphate tablets. (European Journal of Nutrition)
Bioavailability of fortification iron varies widely with the iron
compound used (56), and foods sensitive to color and flavor changes
are usually fortified with water-insoluble iron compounds of low
bioavailability. Iron compounds recommended for food fortification by
the World Health Organization (WHO) (56) include ferrous sulfate,
ferrous fumarate, ferric pyrophosphate, and electrolytic iron powder.
Many cereal foods, however, are fortified with low-cost elemental iron
powders, which are not recommended by WHO (57) and these have even
lower bioavailability (AJCN, 2010).
The results of serum and urine analysis indicated that Mg
bioavailability was comparable for mineral waters with different
mineralization levels, bread, and a dietary supplement. (Tandofline,
Mg supplements comparison:
Studies on the bioavailability of different magnesium salts
consistently demonstrate that organic salts of magnesium (e.g., Mg
citrate) have a higher bioavailability than inorganic salts (e.g., Mg
oxide) (Nutrients, 2019)
The bioavailability of potassium is as high from potatoes as from
potassium gluconate supplements. (AJCN, 2016)
In conclusion, even if most studies mentioned in this answer suggest that nutrients from foods and supplements are equally bioavailable, you need to check specific supplement formulations, for example, iron from many fortified foods and magnesium oxide can have poor bioavailability. Anyway, the studies show that most people with normal blood nutrient levels do not need dietary supplements (Int J. Prev. Med., 2012 ; Annals of Internal Medicine, 2014).