Carbonation issues with bottling

I have a 5 gallon batch of brown porter which has been in bottles for almost 3 weeks and when i check the carbonation the beer is flat, no bubbles or head.

I bottled using 4 oz. of priming sugar which I mixed with 1 cup of water. The beer has been at room temperature in a closet for the entire time.

Should I give this batch a little more time?

Also, I was looking at trying to use the Cooper's Carbonation tablets but am worried it might over-carbonate.

Thoughts?

Did you boil the water and then mix the sugar, and then cool it down, and then add it to the bottling vessel, and then rack beer on top of it? What's room temperature? If you gently shake a bottle, does it fizz, leak, etc?

Also, put the bottle in the fridge for 48 hours before you test it to let Co2 go into solution better.

Just let it sit. It will carbonate if you followed the correct bottling procedure. Some take more time than others. It depends on many factors exactly how long it takes to carbonate. I usually don't try the first one until 4 weeks anymore. Ive had many ready at 3 weeks, but they were better at 4 weeks. Also, the gravity of the beer will also determine the amount of times it takes to carbonate, meaning higher gravity beers might take longer and most of the time need to bottle condition for quite some time anyways.

I could be wrong, but 4oz of sugar for 5 gallons is on the lower side. Also... not sure what your "room temperature", but the rule of thumb I've heard is 1oz of sugar per gallon and leave it at 70F for at least 10 days. If you chill it before you serve it, that can cut down on the amount of bubbles you see as well.