Is GDP biased against public goods?
There is no evidence GDP is biased against public goods. Moreover, that twitter user you reference clearly did not understand what public goods are because the user gives examples of private goods as public goods. What the twitter user was likely talking about, when the user said public goods, are actually government provided goods and services no matter whether they are public or private. However, there is no evidence that GDP is somehow biased against government provided goods.
It is It is true that GDP excludes non-market transaction, but market transaction here does not mean that the transactions are private. For example, spending for soldiers salary in the military (national defense is a public good as it is non-rivalrous and non-excludable) is not done via free-market mechanism, but it is still considered a market transaction for the purposes of calculation of GDP. Market transaction from GDP perspective means some verifiably recorded transaction.
Things that are excluded from GDP can be any goods private or public that are not bought and sold in some way. For example, your home production (e.g. home cooking, house cleaning) is excluded from GDP because there is no verifiable record of much you value that home production. Moreover, note that most of home production, such as cooking is private good (since your home-cooked meal is both excludable and rivalrous).
if goods and services such as healthcare is converted from public to private, GDP will increase, because GDP's value is received from market prices, and public goods do not have a market price as such.
First, this argument is prima facie wrong. Healthcare is not a public good but private good. Healthcare is, generally speaking, both rivalrous and excludable so it is private good. Government provision of a good does not make a good public good. Government can provide both private goods or public goods (e.g. in some countries government provides free food to the poor but the food is still private good as it is excludable and rival). Public goods have to be non-excludable and non-rivalrous (see Mankiw Principles of Economics Ch 11) and most of healthcare, with exception of perhaps things like certain vaccines that not just help prevent individual disease but also reduce spread, clearly isn't either.
Second, even if the person would be talking about government provided goods it is not necessarily correct. Socialized healthcare such as for example NHS in the UK is not free. Government pays for the doctors, medical equipment, medical procedures and medicine. All this is going to be counted toward GDP which records government spending.
If NHS would be converted into private company but they would spend the same amount of money for doctors, medical equipment, medical procedures and medicine GDP would stay same.
For example, the Netherlands has healthcare system based on private insurance yet according to the OECD statistics in 2019 the Netherlands spend exactly the same share of GDP on healthcare as UK (see OECD statistics). This clearly illustrates that even when it comes to healthcare government provided healthcare is not necessarily cheaper. Rather what matters more is how the system and incentives are set up. For example, in the NL government forces private companies to offer a basic healthcare insurance at a regulate prices to anyone, but the system itself is private (i.e. government does not own centralized healthcare insurance like UK does with NHS).
but in reality, I feel that goods in the private market are marked up higher.
I do not think there is much evidence for government generally paying less for goods and services. There are for sure some industries where government can get things cheaper but that is not a general rule.
For example, in early 2000s there was a scandal that revealed that in the US Medicare payed as much as eight times what other organizations payed for the same drugs and medical supplies (see Committee on Appropriations hearing records).
Furthermore, also research by Bandiera, Prat & Valletti (2009), published in American Economic Review (which is one of the best econ journals) shows that governments actually often tend to pay more than other organization for goods and services due to passive waste. That is they often pay more because:
waste can derive from a variety of sources. One is that public officials simply do not possess
the skills to minimize costs. A second is that public officials have no incentive to minimize costs. Another potential cause of passive waste, following Steven Kelman (1990, 2005), is that
excessive regulatory burden may make procurement cumbersome and increase the average price
that the public body pays.
So actually, evidence shows that it is other way around. Government typically pays more for goods and services even though there of course can be exceptions.
My question is are these really problems in measuring GDP, does GDP counter it in any way?
The problems of not measuring non-market transactions, and the problem that higher GDP does not always mean people have more welfare (as sometimes GDP could be higher just because of above mentioned passive waste) are actually valid problems of GDP.
However, it is not valid to say that these bias GDP toward private sector. Rather GDP has bias towards 'official sector' regardless of whether we talk about private or government organizations and regardless whether we talk about private or public goods & services, at an expense of provision of private or public goods & services at grey or black markets that still can (in the case of black market sometimes) provide useful goods and services (e.g. home cooking or in case of black markets drugs like cannabis).
For example, suppose government would prohibit people cooking at home. Such policy would likely raise GDP as people would switch from the grey home production to buying good via official market exchanges. Yet they would not necessarily experience increase in welfare. If government would further nationalize food producers the goods and services would be government provided (through still classified as private goods and services) and it might even increase GDP if there is passive waste but people wont be necessarily better off.
Hence, there is a kernel of truth that GDP is imperfect and a biased measure. However, it is not true that it is biased in favor of private provision of goods. Rather the bias is toward 'official' exchanges at an expense of gray or black economy that still contributes towards the gross output of an economy that GDP tries to measure.