Is there a way to scan through a signal and remove the parts that don't look "normal". This would mean using more advanced then a simple pass band filter, but something that can look at a period in the signal and remove it if that period didn't fit into a "template".
The short answer is no, unless you know exactly what the noise is, you can not in any practical sense remove the noise. There are things you can do to maximize SNR but no technique exists to remove all noise.
You have to analyse your signal and noise spectral distribution. If by chance your signal and noise are not distributed over the same frequencies, you can design a filter ( high pass, low pass or bandpass) depending on those frequencies. If noise occupies the same frequencies than your signal, it will be complicated. If you have a periodic signal, you can always average on multiple acquisitions which will enhance the SNR. In your case, it can be noticed that your noise is pulsed. You can blank some sequences of your signal corresponding to high signal power level (threshold to be defined) in order to remove some noise.