I'm frustrated by these short term contracts and would like some advice. I've been offered several one or two month extensions after my fixed term ended in May. My PI said he may have longer term funding next year but for now he wanted me to stay while waiting for grant results. I wondered if I should continue to live on these short term contracts and wait for the uncertain grant results or start looking for another job? I want to stay in academia.
As a fixed-term postdoc you should pretty much always be looking for another job, and your PI should expect this. Unless you have more than about 18 months left it is worth keeping an eye out for other plausible postdocs - it takes time to go through the process, if you get it they may be flexible about start date, and even if they aren't you have probably gained a year or so of security if the new job is 2 years. If you have longer than that left, is there anything semi-permanent you would be competitive for? It is a bad idea to wait until your postdoc is almost finished to start looking - academic jobs are difficult to get and you should expect to make multiple applications before finding something.
So definitely start looking now, if you haven't already. Of course it is nice to have an extra couple of months to fall back on if you don't get something else, but that's how you should think of them. And you should think of the possibility of your PI getting extra funding as no different to any other potential job you haven't got yet - you need to be pursuing multiple options to give yourself the best chance to get one of them.
My advice would depend on your options. I suggest that if you have no other good options that you continue with this. But simultaneously you can and should look for something more permanent.
A short term situation can help get you over the bridge to something better, but don't neglect the search for that something better. And if you have to leave "at a moment's notice" then be prepared to do that.
I can't say whether it is "normal". It can be a good thing if you work it right, however. Just keep your acceptance contingent on a better offer.
To me, this sounds like your PI is effectively trying to give you a bit of bridge funding to keep you around while they try to get real funding for you.
This is not "normal" in the sense of a desirable situation, but you already left desirable outcomes behind when your position ended without you having either received a new contract from your PI or having obtained a new position for yourself elsewhere.
You need to be searching hard for a new position elsewhere. If your PI happens to land grant money in time to make you an offer for a position, that's great, since it sounds like you'd like to keep working for them. But right now, they don't have it to give you, or else they would already have done so. If you want to stay in academia, you're going to have to start finding your own positions and money, and now is the time to do so.
The only remaining question is whether you are better off staying employed on short-term contracts with your current PI while you search. Having a short gap on a CV won't hurt your search for a new position, but if you're enjoying working with your current PI and doing so won't inhibit your search, there's no reason not to stay employed while possible.
I actually think this is not completely abnormal- with a caveat. This is always a possibility when your PI has some additional funding floating around; however it is in most cases intended to give you a backup when you have applied for other positions but were unsuccessful. In other words, it should be used as a lifeline to give you a few more months to find another position.
If you find yourself here, hopefully it is only because you were applying for lots of other things, and are still waiting for decisions on some of them. In any case, you should be applying for everything as if your future career depends on it (because it does). Do not count on your professor being funded to keep you around, because this is far from guaranteed.
It's sometimes seen as bad form to keep a postdoc past the end of a fixed term contract (excluding the +1 year in the typical 2+1 format of my field). Though I know of labs in adjacent fields where this practice is pretty common.
The general guide is if you want someone around longer, recruit them for a more prestigious position. A 5-year job is more prestigious than 2-year, tenure track is the best, etc. Extracting continuous fixed term work is a little scammy, but it definitely happens. See also: keeping a grad student around forever in TAships so you don't have to pay them a postdoc salary when they are basically at the level of experience of a postdoc, and intimately familiar with your lab (good for you, bad for the student who probably needs more broad experience).
Definitely keep looking for another job.
Is it normal to have several one or two month extensions after a fixed term contract postdoc?
Unfortunately, yes, in many places.
In the research groups in which I have encountered in the USA, UK, and Canada, more than 90% of employees are on temporary contracts. In all of them, the situation you describe happened. I've seen colleagues who were on their 17th contract extension. I've seen colleagues who chained temporary contracts at the same institute for more than 30 years. When project n ends before project n+1 started, either the advisor would find bridge money 1 or 2 months at the time, or colleague would be temporarily unemployed. When an extension happened, sometimes the new contract would come several weeks before the end of the last one. Sometimes several days before. Sometimes after, which either means working for free hoping the promised retroactive pay was going to happen, or filing UB40. If you need a visa with a contract renewal the situation is even worse.
This is my experience at universities, where each professor is more or less their own island. The situation may be a bit better at (government) research labs, but I'm not sufficiently familiar with those to comment on it specifically.
It's the fate of those who stay in Academia and are good enough to produce good research but don't manage to get tenure. I've personally managed to avoid this by securing my next post-doc well before my previous one finished, which has had the effect that I've never actually finished a post-doc to the nominal ending date; no one has blamed me for this in the slightest.
Some countries have laws or policies protecting how long one can be on temporary contracts or how many temporary contracts can be chained. In those, such researchers will be either forced out of academia or forced out of the country. If you find yourself on your 10th short contract, you may want to think about alternatives to academia — or not, if you're happy to accept the perpetual uncertainty. This is academia in 2020.