New to brewing - same underlying taste in 4 batches

I have now produced 4 batches, all 5 gallon extract on the kitchen stove. 2 were kits; an American Light and a blond ale. 2 were recipes I made, variations of beersmith recipes that I had to substitute hops because brew store supply. Although they all had a different flavor, every batch had an underlying taste which made them feel heavy. In batch 3 and 4, i made a yeast starter, thinking this would create a crisper finish, it didn't. The routine has been one week in fermenter, two weeks in carboy, two weeks in bottle. Each time, I end up letting them sit another 2 or 3 weeks in bottle just to let brew mellow out.
My buddy brews all-grain and his brew is clean and crisp, with a distinctive intended flavor. Not sure what to do different to get rid of the heavy mouth feel. Any thoughts?

Let them sit longer in the primary. It helps if your beer can sit for 3 weeks or so on the yeast cake so the yeasts can consume fusel alcohols and diacetyl left over from the primary fermentation.

Second, use a yeast with good attenuation that is also highly flocculant (Ex: WLP001) and cold crash the beer for 48 hours prior to bottling to settle out suspended yeasts.

Third, if you can control your fermentation temperatures using a temperature controlled fridge or freezer, use lager yeast and lager the beer.

My first few batches with LME had the extract "twang" that you read about -- not sure if this is what you are referring too. I switched to DME and it got noticeably better, then I switched to all grain and the beer has been amazing since.

Another advantage of using DME over LME is shelf life. LME goes stale pretty quick.

Give them longer in primary, skip secondary (for anything using ale yeast, if it's not a BIG beer) and don't rush things. IF you're fermenting at the right temperature range for the yeast, for long enough, you'll get better beer. Give them 3+ weeks at 70F once in bottles to carbonate, then 5-7 days in the fridge before pouring one to glass.

IMO, you would be better off starting with 2-4 weeks in primary before bottling. Make sure the starter is of the correct size for the batch (use yeastcalc.com to find out how big by batch size, OG, yeast production info an aeration method). Oxygenate/aerate the wort to the proper level also makes a huge difference.

BTW, there's already tons of threads covering all of these aspects.

I am new to brewing as well and I am 6 batches deep. My first batch was pretty mediocre as it was rushed and the ingredients were quite old. Batches 2 and 3 tasted great with no weird off flavors at all. 4 is conditioning now and 5 and 6 are still in the primary. I use DME and LME for now until I build a mash tun. I brew, ferment in a swamp cooler @ 65-68F for 3+ weeks, bottle and condition. I try my first beer after 10 days and second after 21, if the beer is tasty I will drink it up except for about 6 or so that I let sit to age. I think your problem is having only a week in your primary, no need to secondary IMO for ales.

Another thing you can do is add half the extract at the end of the boil instead of all of it for the full 60 minutes. This usually eliminates that "twang" and gives a truer color to the beer.

All other suggestions are spot on! Happy new year and here's to better beer!