Sales of newly built homes rebounded and prices rose briskly in April, signs of rising demand in a housing market constrained by tight supplies.

New-home sales climbed 6.8% from March to a seasonally adjusted annual rate of 517,000, the Commerce Department said Tuesday. The median price of a new home sold in April stood at $297,300, up 8.3% from a year earlier.

A separate measure of home prices rose for the 35th straight month. The Standard & Poor’s/Case-Shiller Home Price Index rose 4.1% in the 12 months ended in March.

Steady job growth, historically low interest rates and rising rental costs have been pushing potential home buyers into the market. But with home prices rising faster than incomes, some buyers may be squeezed out of the market. Supplies of existing homes are tight and home builders have been slow to start construction.

The crosscurrents have been reflected in other recent housing data, which showed a drop in existing-home sales last month alongside more strong price gains.

“Seeing new homes become more affordable for new buyers is a really important part of getting the housing market back to normal,” said Stan Humphries, chief economist at real-estate firm Zillow.

Mr. Humphries and other economists believe the U.S. needs construction of at least 1.5 million new homes a year to keep pace with a growing population and the normal loss of existing stock due to age.

So far this year, builders are averaging just over 1 million starts, not much different from 2014. The pace picked up some in April, hitting the best level since November 2007.

Woodside Homes, a closely held builder operating in five western and southern states, sold 453 homes in the first quarter, up 30% from a year earlier. Woodside Chief ExecutiveJoel Shine described spring sales as “reasonably strong” rather than spectacular. He noted that job growth has remained strong and interest rates remain below 4%, with the expectation of an interest-rate increase later this year providing some buyers with extra motivation to buy now.

“It’s sort of been the recovery that never has gotten spectacular but doesn’t go away either,” Mr. Shine said. “Little by little, the economic trends and demographics trends are going our way.”

So far this year, monthly sales of new homes have averaged 515,000, a pace that if continued would mark the best since the first half of 2008. Over the past year, new-home purchases are up 26.1%.

The market in April had 4.8 months of supply of new homes, reflecting how long it would take to exhaust existing inventory at the April sales pace. That was well short of the 6-month mark many analysts consider a balanced market.

New-home sales reflect about one-tenth of all home purchases, and monthly estimates are often heavily revised as more data become available. The April figure has a margin for error of plus or minus 15.8%.

While the Case-Shiller index also showed prices rising, the pace has slowed since last year. That trend may continue.

“Home prices are currently rising more quickly than either per capita personal income…or wages…narrowing the pool of future home-buyers,” said David Blitzer, managing director and chairman of the Index Committee for S&P Dow Jones Indices. “All of this suggests that some future moderation in home prices gains is likely.”

New-home sales peaked in July 2005, when they hit an annual pace of nearly 1.4 million. Sales bottomed out at 270,000 in February 2011 as the housing market crashed.

Figures so far from 2015 put sales closer to—but still short of—historical averages. From 1990 to 2000, for example, monthly new-home sales averaged 711,000.

- The Wall Street Journal