The title of this piece is actually a cheat. It was selected because it is more catchy than "Is Science Scientific?". So, with that out of the way let me address the question.
There are two main pillars to Science: data and mathematics. Scientists collect data then they organize that data using mathematics and related tools into theories. If there is a problem with data or if there is a problem with mathematics, Science is in trouble. So let's look at each separately.
Data is, well, data. But is it really? How do we know that the data Scientists collect is true? Scientists have a lot of reasons to believe their data but it is all "handwavium". "Handwavium" is shorthand for a hand waving argument, an argument that is convincing to someone who already believes but is unconvincing to an opponent. I won't review them. Instead I will note that all "data" is filtered through the human nervous system. The human nervous system modifies things in lots of ways. Then it delivers it to the brain. Frankly, we don't understand what happens in the brain very well. So how do we know the result is "truth"? We don't. There are various properly done versions of this argument but instead I will refer you to the movie "The Matrix". The conceit of this movie is that for reasons that don't actually make any sense a whole system has been set up to harvest energy from humans. To make the system work computers control the nervous signals to humans and impose an artificial reality on humans. Taken to an extreme there is really no way to prove that this is not happening to all of us in the "real world". So that there is no Scientific way to prove that scientific data (or data of any kind) is true.
So there is a problem with data. It turns out that there is also a problem with mathematics. There is a long history of trying to make mathematics mathematical. This undertaking culminated with a 1910 book called "Principia Mathematica" by Whitehead and Russell. The title referred to the famous book with a similar name by Sir Isac Newton. Whitehead and Russell attempted to do a proper job laying out the foundations of mathematics. But people kept finding small problems with the work. For some time most people were of the opinion that the problems could be fixed and eventually the endeavor would be a success. Then in 1931 Kurt Godel published a paper that destroyed the whole thing. What Godel showed was that for any "sufficiently complex" mathematical system there would always be propositions that could neither be proved true nor false. The most well known example of this is the statement "this statement is false". If we assume the statement is false, then it is true. If we assume the statement is true then it is false. Thus it is neither true nor false. So the whole project to put mathematics in a "firm mathematical foundation" is doomed to failure. In short, mathematics can not be Scientifically proved to be true.
So we end up with the situation where the two pillars of Science are not Scientific. So Science is not Scientific. So all those people braying that "Science is just another type of religion" are right. Is all lost? In my opinion, no. Science ultimately does depend on belief. But there are pragmatic reasons to believe that Science is a dependable method for arriving at truth. But, before going there let me make a digression.
I imagine a "truth" scale. It has five regions. At one end is the set of "things that are true and we can prove them to be true". This is balanced by a region at the other end of the scale consisting of "things that are false and we can prove them to be false". Back at the other end is a region more toward the center consisting of "things we believe to be true but can't prove to be true". This is balanced at the other end of the scale by the region of "things we think are false but can't prove they are false". And, of course, in the center we have the region of "things where don't know if they are true or false".
We would like for Science to be in the "things that are true and we can prove to be true" region. But as I have shown above, Science is in the next region, the "things we believe to be true but can't prove" region. It seems that at this point we haven't made any real progress but, in fact, we have. If Science can't reliably show things to be "provably true" it turns out that Science is really good at proving things to be "provably false". For instance, Science puts "Biblical Creationism" solidly into the "provably false" region.
So now let me get back to why it is reasonable to believe in Science even though we can't prove it is true. Science has been demonstrated to deliver reliable truth over and over. A simple example is cell phones, and technology in general. Cell phones work because something called quantum mechanics works. Quantum mechanics is really weird. It is literally unbelievable and unnatural. It is also extremely complicated. But it works. We know it works because cell phones work. There are very few true believers who do not own and use a cell phone. Ask them if they believe in cell phones. They don't understand the question. But if they did they would be forced to say "yes". We know they would because they depend on them on a daily basis. If you believe in cell phones you must believe in quantum mechanics. Every time you get on an airplane you are putting yourself in a position where your life literally depends on Science being true. The same is true for cars and many other aspects of the modern world. Cell phones that work, airplanes, cars, the whole lot of it, is proof playing out before our eyes that there is truth to be found in Science.
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