Mortgage interest rates have turned upward again, pushing thousands of American families back out of the home-buying market and threatening to undermine the fragile housing recovery.

The upturn comes after almost a year of steady improvement, which had cut rates on conventional fixed-rate mortgages from more than 17 percent to less than 13. So far the rise has not been large; rates hovered between 12 3/4 and 13 3/4 percent for conventional loans at week's end, but there were several indications of trouble:

* The Federal Home Loan Bank Board reported this week that the average interest rate charged for new mortgages rose to 13.01 percent in June, the first increase since July 1982 when home mortgage rates were 17.22 percent. "A general rise in market rates which began in mid-May" touched off the increase in home loan rates, the board said.

* The number of "points" charged to buyers and sellers of FHA loans increased to between seven and eight in the Washington area and as many as 10 in some parts of the country. Each point represents one percent of the mortgage amount, and when the numbers reach this week's level, home buying is choked off, say industry analysts.

* Pressure rose also in the financial markets as Federal Reserve sources said the Fed, fearing another inflationary spiral, will tighten credit and let interest rates rise.

The interest rate on FHA- and VA-insured home loans was cut two months ago to 11 1/2 percent, its lowest level in nearly three years. But when the market rates on conventional loans failed to follow, HUD pushed its rate back up to 12 percent in early June.

"We've probably seen the best we're going to see on mortgage rates for several months to come, if not for the whole year," said James Christian, chief economist of the U.S. League of Savings Institutions. Christian added that the future of interest rates "depends so much on what Congress does about the 1984 budget."

If interest rates continue to rise, "It will mean a downturn in housing starts and in sales," said Harry Pryde, president of the National Association of Home Builders. "It is important to continue to have a sustained recovery in housing and in the country as a whole."

The rising rates and rising number of discount points can create chaos for home buyers because lenders cancel loan commitments and renegotiate when the interest rates go up, said Mark Riedy, executive vice president of the Mortgage Bankers Association. "It's disruptive, and what happens is that sellers go out of town, get sick, can't get to closing, while they're waiting for the rates to change," Riedy said.

The plight of one prospective home buyer in Washington illustrates the troubles that buyers face. He applied for a VA loan in mid-May when the interest rate was 11 1/2 percent, and four points were being charged.

The FHA requires that buyers pay no more than one point; sellers must pay the rest.

"I was shocked to find a month later that the rate was up to 12 percent," said the Washington man. The points, which had risen to around 6 and 7 during the previous weeks, came down to 4 1/2, he said. Within one or two weeks, the interest rate was still 12 percent and the points were back up at 7.

The man's lender has told them that "they won't lock in the rate until we go to settlement, and they say they cannot settle till they have the appraisal on the property, and that they haven't received it," the man said. The appraiser visited the property about three weeks ago, and the buyer says he's "getting suspicious" that the delay in going to settlement really is caused by the seller or lender foot-dragging until the rate is changed.

Similar problems are being experienced by families all over the country, said Felix Beck, head of Margaretten & Co, Inc., a New Jersey mortgage banking firm and first vice president of the Mortrgage Bankers Association.

"Sellers are unwilling to go into deals where they must not only pay the real estate commission but also must pay a significant number of points," he said. The result is that "the deal is in jeopardy, and you have situations where people virtually have their furniture on the moving trucks when they find they can't conclude their deals."

Beck said the home mortgage market "is highly unusual and highly disruptive."

The crunch was putting heavy pressure on the FHA this past week to boost its interest ceiling by at least half a point, and officials of the Mortgage Bankers said that a boost of a whole point--to 13 percent--is what is really needed. But even that still would force lenders to charge three or four points, they said.

Most mortgage loans today--particularly those made by mortgage bankers, who do the bulk of FHA and VA leanding--are sold to investors in the so-called secondary mortgage market. In order to make these loans competitive with other available investments, lenders must discount them to raise their yields. The farther below the market rate the FHA rate is, the deeper the discount the lender must make. The lender then recovers his discount from the borrower in the form of points.