Financing U.S. government budget deficits, and their impact on interest rates
The treasury can issue more bonds and the central bank buys it up, prints more currency and gives it to the government. This is called monetizing the deficit. 
Although this practice may be challenged by the central bank itself depending upon the situation.
The treasury can sell new bonds to commercial banks and other financial institutions. Commercial banks use these to be kept as statutory reserve requirements with the central bank and also as safe assets because these are, of course, most highly rated securities. The demand could be more or less but there is always a demand for safe assets. Think of a countries like China and Japan which invest heavily in U.S treasury bonds and other assets. Theoretically speaking, government can finance any amount of deficits via the central bank route. 

Crowding out will happen when the government is borrowing huge loans from the commercial banks and dries up the sources of funds for the private investors. It also depends on the purpose and timing of the spending. Whether the government is spending on non-productive programs or productive programs which are in turn helping the overall health and investment environment of the nation and whether it is spending counter-cyclically or cyclically. Also, if there is an upward pressure on the interest rates due to possible crowding out, we can expect central bank to make necessary adjustments. 
Moreover, the mechanism is not so simple. It is complex as hell. The flow of savings in the economy is not fixed and hence the pressure may or may not build up. It largely depends on the investor sentiments and expectations also.