Tax cuts don't increase debt. Out of control spending by the U.S. Congress is what is driving the debt.
Sovereign or national debt is made up of two parts.
1. Debt owed held by the public. The public are those that purchase it's bonds, domestic and foreign purchasers.
2.intragovernmental debt or debt it owes to other government departments. For example, the U.S. government is raiding American citizens Social Security retirement accounts for funds. SS retirement has been paid for by taxes from American's pay cheques. Supposedly, it would be paid by with interest upon retirement in monthly payments. The money will soon run out because the government has been stealing it to fund other things.