This is EXACTLY the reason why internet genealogy doesn't help........
How have you got back to where you are now? You mention lots of websites and bmd 'records' however if you have only ever typed in a name, clicked and up pops information which you have copied then sorry but you will hit brick walls when the website runs out of databases to draw from. NO website has all records and never will, you need to step away from your computer and look at records.........
This situation is why you never start research online you start at home with records you already have there, which gets you back for free 3-5 generations, HOWEVER and far more important it teaches you what records are and makes it easier for you to recognise, problem solve and so you avoid brickwalls, it teaches you what is available and we are all learning about that all the time, after 30 years of researching I am still learning about records which are available to me ( but they are not online)
Number one stop being 'desperate'................ for you this is a hobby, not a life/death situation.....
Number two realise that genealogy is like a jigsaw puzzle, you research to find the pieces of the picture which is slowly built up and often it is like a jigsaw which is no longer in the box so we have no picture guidelines to follow..........
Number three genealogy is NOT a competition about how far we get back,but about quality of information.
............and remember ONE mistake and you no longer have YOUR ancestors in your tree...... so make sure each generation is correct before you jump to the next, or you will end up with a collection of unrelated names and hit brickwalls, at this point many people give up or they continue to defend their collected information get desperate and just copy and paste anything they can find regardless of if they know it is right or not just to continue to search.
It took me 15 years to find the information about ONE man, my great grandfather and with that line I couldn't go further until I had all his information, yet on his lines I am back now with full recorded cited records to the 15th century.
You need to be clinical, and unless you are you will find you hit this point each generation so look at what you do know and that is ONLY what you can PROVE.
Prove= a REAL record, which you have a copy of or have seen and can cite, so the ONLY research you have done online is when you have seen an image of the record NOTHING else is research, it is just copy and paste information which you can't prove.
You say your 4th great grandfather was born in 1815, so what RECORD PROOF do you have of that?
As you haven't found his baptism record or you would know his parents names and the baptism record is the obvious record you need to find..... you say it is NY, so where in NY and what religion was he, so which church records do you need to look at.....
Have you found his marriage banns, as his parents may be written on that.
I am NOT a fan of familysearch, however many organisations around the World have documented, transcribed and scanned images of records and given then to the LDS under a contract which requires them to offer them for free to the public and so this is a resource IF it is an image and a CLUE to where to look if it is only a transcription...so if a transcription says "baptised on 25 April 1815 at PRESBYTERIAN CHURCH,MONTICELLO,SULLIVAN,NEW YORK" then you need to look at images or the real records for that church and make sure the transcription is correct ( as many of familysearch transcriptions are just IGI/collections and are not correct)
https://familysearch.org/search/collection/list#page=1&countryId=22&dateRange=dateRange4 some information that might help you narrow things down and when you get all the information you can from that side of the pond and get everything you can before you attempt to jump to my side of the pond and start looking in Ireland as if you think it is hard work now, it is even harder for someone who doesn't know another countries record system...then and ONLY then this may help you, which is an Irish/UK based website http://familytimeline.webs.com/