Here are the results recorded by Byron Nelson in PGA Tour events in 1945, when he won 18 of 30 tournaments entered, including 11 consecutively. The full, tournament-by-tournament results listed at the bottom are based on records provided by Jon Bradley, Mr. Nelson's longtime agent.

But before we get to the full record, let's start with ...
11 In A Row

Here are the 11 tournaments that make up the most-famous winning streak in golf, along with Nelson's margin of victory in each:

• Miami International Four-Ball (team tournament)
• Charlotte Open, won by 4 strokes
• Greater Greensboro Open, won by 8 strokes
• Durham Open, won by 5 strokes
• Atlanta Open, won by 9 strokes
• Montreal Open, won by 10 strokes
• Philadelphia Inquirer, won by 2 strokes
• Chicago Victory National Open, won by 7 strokes
• PGA Championship (match play)
• Tam O'Shanter Open, won by 11 strokes
• Canadian Open, won by 4 strokes

A couple notes:

Jug McSpaden was Nelson's partner in the Miami Four-Ball. That's only appropriate, since McSpaden set a couple of his own records in 1945: He finished in the Top 10 31 times, and finished second 13 times (most of those to Nelson).
The PGA Championship was the only major championship played in 1945, due to World War II. Nelson defeated Sam Byrd 4-and-3 in the 36-hole championship match.

18 Wins

Here is the list of all 18 of Nelson's PGA Tour victories in 1945, with his winning score:

Phoenix Open, 274
Corpus Christi Open, 264
New Orleans Open, 284
Miami International Four-Ball (team tournament)
Charlotte Open, 272
Greater Greensboro Open, 271
Durham Open, 276
Atlanta Open, 263
Montreal Open, 268
Philadelphia Inquirer, 269
Chicago Victory National Open, 275
PGA Championship (match play)
Tam O'Shanter Open, 269
Canadian Open, 280
Knoxville Invitational, 276
Esmeralda Open, 266
Seattle Open, 259
Glen Garden Open, 273

A couple notes:

The last win in Nelson's 11-tournament victory streak was the Canadian Open. The streak ended the next week in Memphis where Nelson finished fourth.
The last two of Nelson's wins came in his last two tournaments of the year. He opened 1946 with two straight wins, for a four-tournament winning streak. Only four golfers in PGA Tour history have winning streaks of four or more, and Nelson did it twice (Ben Hogan also did it twice and Jackie Burke once; Tiger Woods has three such streaks - see longest PGA Tour win streaks).
The 259 Nelson posted in winning in Seattle set a 72-hole scoring record on the PGA Tour. It broke a record that Hogan had established just two weeks earlier. Nelson's scoring record stood for 10 years, until Mike Souchak's famous 257 at the 1955 Texas Open. Other than Souchak, nobody bettered Nelson's 259 on the PGA Tour until 1989.

The Complete Record

Below are Byron Nelson's results in all 30 official tournaments he played in 1945. I say "official" because Nelson actually played a 31st event, and he won it. It would give him 19 wins and 12 wins in a row ... except that the event was scheduled for only 36 holes and so counts as an unofficial win.

In addition to Nelson's 18 wins, you'll noticed that he also finished second seven times and never outside the Top 10. Nelson's average margin of victory was nearly 7 shots. Of his 112 stroke-play rounds, 92 of them were below par. He had more rounds below 65 than he had above 72.

Byron Nelson's 1945 Tournament Results

Jan. 5-8, Los Angeles: 71-72-70-71--284, second
Jan. 12-14, Phoenix Open: 68-65-72-69--274, first, won by 2
Jan. 18-21, Tucson: 67-68-67-67--269, second
Jan. 26-28, Texas Open: 67-66-68-68--269, second
Feb. 1-4, Corpus Christi Open: 66-63-65-70--264, first, won by 4
Feb. 9-11, New Orleans Open: 70-70-73-71--284, first, won by 5
Feb. 16-18, Gulfport: 69-68-72-66--275, second
Feb. 23-25, Pensacola: 69-69-71-65--274, second
March 1-4, Jacksonville: 68-66-72-69--275, sixth
March 8-11, Miami International Four-Ball: first (team tournament)
March 16-19, Charlotte Open: 70-68-66-68--272, first, won by 4
March 23-25, Greater Greensboro Open: 70-67-68-66--271, first, won by 8
March 30-April 1, Durham Open: 71-69-71-65--276, first, won by 5
April 5-8, Atlanta Open: 64-69-65-65--263, first, won by 9
June 7-10, Montreal Open: 63-68-69-68--268, first, won by 10
June 14-17, Philadelphia Inquirer: 68-68-70-63--269, first, won by 2
June 29-July 1, Chicago Victory National Open: 69-68-68-70--275, first, won by 7
July 9-15, PGA Championship: first (match play)
July 26-29, Tam O'Shanter Open: 66-68-68-67--269, first, won by 11
Aug. 2-4, Canadian Open: 68-72-72-68--280, first, won by 4
Aug. 16-19, Memphis: 69-73-66-68--276, fourth
Aug. 23-26, Knoxville Invitational: 67-69-73-67--276, first, won by 10
Aug. 31-Sept. 3, Nashville: 70-64-67-68--269, second
Sept. 6-9, Dallas: 72-70-71-68--281, third
Sept. 13-16, Tulsa: 73-69-75-71--288, fourth
Sept. 19-23, Esmeralda Open: 66-66-70-64--266, first, won by 7
Sept. 27-30, Portland: 71-71-67-66--275, second
Oct. 4-7, Tacoma: 70-69-73-71--283, ninth
Oct. 11-14, Seattle Open: 62-68-63-66--259, first, won by 13
Dec. 14-16, Glen Garden Open: 72-65-66-70--273, first, won by 8

Nelson's scoring average in 1945 was 68.34. That's unadjusted (today the Vardon Trophy is awarded based on adjusted scoring average). Nelson didn't win the Vardon Trophy, however, because it wasn't awarded from 1942-46.

However, only one time in tour history has Nelson's 1945 unadjusted scoring average been bettered. That was Tiger Woods' 68.17 in 2000.