Some remarks:

1) It is not the data or dependent variable that has to be normally distributed, but the residuals! In case of a factorial ANOVA this can also be tested by looking at the distribution of the DV for each group, in your case all 2X5=10 groups should be inspected, not the dependent variable itself.

2) How did you detect violation of normality? Typically a statistical test is used, such as Kolmogrov-Smirnov-Lilliefors or Shapiro-Wilk. But all of these inferential statistics have the power problem: for small sample sizes, where a violation of normality is more severe, these test are unable to find violation(unless really substantial), for large samples, where violations do not matter so much anymore (unless substantial), they will be significant all the time. So, there is not much use of these tests. Have a look at histograms and QQ/PP plots to estimate the violations, and how it is violated (outliers, skewed distribution etc.).

3) Ouorou Ganni Mariel Guera is right that transformations may help sometimes, but that they will distort the interpretability of your results. Therefore, do it with care. If you really want to do Box-Cox in SPSS, you can find it via the extension hub (to use some selected R programs), where you can install the RegBoxCoxTransforms package to do it in SPSS.

4) According to the central limit theorem, the distribution of the means will tend to be normally distributed with increasing N, as Ouorou Ganni Mariel Guera already said, but it is not guaranteed, especially if you have outliers, which drive the effect. It is not necessary to switch immediately to non-parametric tests, there are a lot of robust estimation techniques you can try out. Please have a look at the books by Rand R. Wilcox, where he describes lots of methods (all in R, but with code or his own R package).

5) A more basic question to start with: what kind of dependent variable do you have? For example, count data or bounded data (like reaction times) are typically not suiteable for typical ordinary least squares methods like ANOVA or regression, since the afore mentioned normality of residuals assumtions, which will be most often violated with such data. For example, if you have count data, which are bounded by 0 and have only integers, a Poisson regression whithin the generalized linear models is more suiteable.

5) My question: why did you construct age groups, instead of using it as a metric predictor in a regression model?

You should not use parametric tests if your data does not meet the normality assumption. Logarithmic transformation, despite being the most used, is not the only one. In fact, it is part of a family of transformations called Box-Cox transformations.

I fear that the Box-Cox transformation is not available in SPSS. To do it in R, this link will be useful: https://www.rdocumentation.org/packages/EnvStats/versions/2.3.1/topics/boxcox ).

These transformations group from the most basic to the most complex. I recommend the use of the basics because too much transformation denatures our data and distances us from the reality of these. If with these transformations you do not solve, you will have to consider non-parametric statistics (although this does not give you many options for the type of analysis you want to perform). Also keep in mind that increasing the sample size (n) increases the probability of obtaining normality. So, if you are working with a small sample it will be more difficult to attend to that assumption.

Regards !

Some remarks:

1) It is not the data or dependent variable that has to be normally distributed, but the residuals! In case of a factorial ANOVA this can also be tested by looking at the distribution of the DV for each group, in your case all 2X5=10 groups should be inspected, not the dependent variable itself.

2) How did you detect violation of normality? Typically a statistical test is used, such as Kolmogrov-Smirnov-Lilliefors or Shapiro-Wilk. But all of these inferential statistics have the power problem: for small sample sizes, where a violation of normality is more severe, these test are unable to find violation(unless really substantial), for large samples, where violations do not matter so much anymore (unless substantial), they will be significant all the time. So, there is not much use of these tests. Have a look at histograms and QQ/PP plots to estimate the violations, and how it is violated (outliers, skewed distribution etc.).

3) Ouorou Ganni Mariel Guera is right that transformations may help sometimes, but that they will distort the interpretability of your results. Therefore, do it with care. If you really want to do Box-Cox in SPSS, you can find it via the extension hub (to use some selected R programs), where you can install the RegBoxCoxTransforms package to do it in SPSS.

4) According to the central limit theorem, the distribution of the means will tend to be normally distributed with increasing N, as Ouorou Ganni Mariel Guera already said, but it is not guaranteed, especially if you have outliers, which drive the effect. It is not necessary to switch immediately to non-parametric tests, there are a lot of robust estimation techniques you can try out. Please have a look at the books by Rand R. Wilcox, where he describes lots of methods (all in R, but with code or his own R package).

5) A more basic question to start with: what kind of dependent variable do you have? For example, count data or bounded data (like reaction times) are typically not suiteable for typical ordinary least squares methods like ANOVA or regression, since the afore mentioned normality of residuals assumtions, which will be most often violated with such data. For example, if you have count data, which are bounded by 0 and have only integers, a Poisson regression whithin the generalized linear models is more suiteable.

5) My question: why did you construct age groups, instead of using it as a metric predictor in a regression model?

Excellent answer by Rainer Duesing . The only thing I'll add is that if you do decide to go with a nonparametric approach, the aligned ranks transformation anova (ART) is a good way to go with a factorial design. (See: http://depts.washington.edu/ilab/proj/art/index.html or http://rcompanion.org/handbook/F_16.html [Caveat, I am the author of the second page.].)

Good remarks, @Rainer Duesing. Thank you.

A permutations test can give you an adequate solution. Look at chapter 4 of the book by Phillip Good (Permutation tests, Springer Verlag, 1994, I think there is a newer edition)